I am not sure that I understand everything that you are saying but if Black knew in advance that the White was going to take on c6 then just about any developing move would be better than 3. ...a6. 3. ...a6 is played with the assumption that the White is (probably) not going to exchange on c6.
Berlin Defence > Morphy Defence for Mainline Ruy?
Ajedrecito is right. If 1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. O-O a6, I was of course wrong to say that white's two choices - Ba4 and Bxc6 - simply transpose into Morphy Defence continuations. Ba4, which, according to Chess365, has been the more popular choice, albeit with small sample sizes (148 vs. 80) does indeed transpose into the mainline Morphy. Bxc6, however, does not, and, as far as I can tell, it is this move, and only this move, that justifies 3...a6 as the best alternative for a mainline black Ruy. An early theme in the Ruy is that white would like a bishop-for-knight exchange followed by a taking of the unguarded e-pawn, but can't because of the threat of Qd4, forking the white knight and pawn, followed by Qxe4+, gaining the material back and ruining white's position. But once white castles, the threat is real. Black's choices are to either get the king's bishop out of the way to prepare for castling (ideally while actually guarding the king - Be7) or to take the material first, giving himself time to get out of the way before the knight is forked in front of the king by the rook. Any alternative, and Nxe5 can't be avenged. So, if 1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. O-O a6, white can respond 5. BxNc6 dxBc6 6. Nxe5, and he should.
Pretty basic stuff I'm sure a lot of people are thinking, but apparently not for the 148 people playing in tournaments who let black transpose into a mainline Morphy, and obviously not for me. Well, I learned something at least
Responding to Atos, I agree, but for a fleeting moment I thought it was possible to pre-emptively play a good response to the Exchange Variation without forfeiting the benefits of a6 in the mainline, non-exchange line. I was wrong!

I once used to regularly play an opponent who would always exchange on c6. At first, I played 3...a6 but then I realized that he was going to exchange on c6 whatever I play so I started to play Nf6 or Bc5 to gain on development and it worked. Otherwise though, I stick to 3. ...a6.
I'd typed up a long spiel about the removal of the guard against Black's e-pawn and White's potential e-file pin after O-O perfectly explaining the first 7-8 moves of the Ruy Lopez. As soon as White's e-pawn is defended, the threat against Black's e-pawn becomes real.
Then, I hit refresh and found this had already been discussed about would be terribly redundant.
In summation: As Black, if you're going to play ... a6 against the Ruy, it's best that it be done on move three. As White, when Black leaves book early, always calculate that removal of the guard trick.
I'd typed up a long spiel about the removal of the guard against Black's e-pawn and White's potential e-file pin after O-O perfectly explaining the first 7-8 moves of the Ruy Lopez. As soon as White's e-pawn is defended, the threat against Black's e-pawn becomes real.
Then, I hit refresh and found this had already been discussed about would be terribly redundant.
In summation: As Black, if you're going to play ... a6 against the Ruy, it's best that it be done on move three. As White, when Black leaves book early, always calculate that removal of the guard trick.
Fishzilla, if you've already got it typed up and didn't delete it, you're more than welcome to post it in here; even if it proves to be 100% redundant, it's interesting what can be learned by seeing the same ideas from a different perspective. Otherwise, apologies for wasting the time you spent on it.
Wow, fezzik's post got me reviewing the 2000 Kasparov-Kramnik games with Kramnik using the Berlin as black and those were indeed some brutal and drawish affairs. Actually, in one of them, the positions were so cramped that both sides agreed to a draw despite each still having seven pawns, both rooks, and three out of four of the bishops and knights, which I'm not sure I've seen before!

Here is paraphrase of what John Cox says in his book on the Berlin Wall.
The reason 3..a6 is played, is so Black can quickly rid himself of the pin on his c6-knight. The only variations where Black doesn't benefit from having ...a6 Ba4 played are those in which the bishop is some point attacked on b5.
The opening moves of the Berlin Wall are: 1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. O-O Nxe4 5.d4 Nd6 which attacks the Bishop on b5.

BTW your assumption that when Black gets around to playing ...a6 that White has only two options Ba4 or Bxc6, is wrong. For instance: 1.e4 e5 2. Nf3 Nc6 3.Bb5 Nge7
4. O-O a6 5. Bc4 when Black must watch out for Nf3-g5 ideas. There are other examples where White moves the Bishop back along the f1-a6 diagonal.
"if there were something inherently bad about black having his king's knight at f6 in the Exchange Variation, then we would expect white to habitually punish the Berlin Defence with BxNc6, whereas this is still a minor variation"
I was looking at the Exchange the other day and wondering this very same thing myself. White sometimes plays the Delayed Exchange, encouraging Black to play Nf6 before exchanging. The reasoning being that Black's best move, defending the e-pawn with f6, is no longer possible. So, shouldn't BxNc6 against the Berlin be good?
vs.

by carl: Both have been played and the delayed exchange was played several times by Flor. They are both playable, but White's results are nothing to write home about. Cox has an example in his book on the Berlin Wall. I'll give the moves, but he has a copyright on the notes so you have to buy the book to read those.

the point of a6 is to be able to folow up with b5 WHEN NEEDED (also after the bishop retreats it cannot come back to c4 d3 or e2 now b3 is its only mobility) now for exchange variation the berlin dosent really have an instant exchange variation after a6 the only reason white evin thinks about exchangeing is because it wins him a tempo if the berlin is played playing Bxc6 is out of the question till move 7
what im trying to say i guess is black plays a6 giving white a chance to play an inferior but still verry playable move where if he played Nf6 Bxc6 is out of the question.
Something occurred to me today that seems like it can't be right, yet I can't for the life of me pinpoint why it's wrong. Hopefully someone else can...
It's well known that, for black facing 1. e4 e5 2. Nf3 Nc6 3. Bb5, the most popular defense by far is 3...a6 - by Chess365's stats, it's over five times as popular as the next choice, 3...Nf6 (the Berlin Defence). As I've always understood it, this is because a6 gives black another defensive tool for his arsenal (and some very minor development) free of charge - white's no better off with his bishop at a4 than at b5, and black can now snuff out any lingering bishop threat at an opportune time with b5. So far so good.
Here's the rub. After the Berlin Defence, white finds himself in substantially the same position as he does in the mainline Morphy Defence (3...a6 4. Ba4 Nf6) - black's e-pawn is still guarded by the unpinned knight and his own e-pawn is now under attack. Just as white's most popular response in the Morphy is O-O, white's most popular choice in the Berlin is also O-O. In the Berlin, after 4. O-O, black might (and often does) take the e-pawn for an Open Ruy, and this does often lead to a different game than the Open Ruy after a6. But black also has the option with his fourth move to now play a6. White's only decent choices - 5. Ba4 or 5. BxNc6 - simply transpose into continuations of the Morphy Defence. Ergo, unless most white players facing the Berlin are missing out on a way to exploit it with something other than 4. O-O, it would seem that, for black planning a mainline Ruy, a6 is at least no better than Nf6, since a6 is still a forcing option for black after Nf6. [You can argue that 3...a6 gives a better Open Ruy for black, but the point is moot if black can play 3...Nf6 4. O-O a6 5. Ba4 Nxe4 anyway].
Now here's the real rub - while the Exchange Variation of the Ruy is not nearly as popular as the mainline, it is certainly a possibility. In the exchange, had black played 3...Nf6 instead of 3...a6, he would have been better developed afterward - surely, there is no way a one-square development of the rook's pawn is more auspicious so early in the game than a knight development (and, indeed, if there were something inherently bad about black having his king's knight at f6 in the Exchange Variation, then we would expect white to habitually punish the Berlin Defence with BxNc6, whereas this is still a minor variation).
So, unless I'm wrong, the massive popularity of 3...a6 is a mistake, albeit a minor one. a6 is certainly better for certain non-mainline options like an early d6, but the fact that Modern Steinitz > Old Steinitz can't possibly account for a6's popularity - it's undeniable that most black players planning a mainline Nf6 are prefacing it with a6. In doing so, they seem to be missing out on a minor hedge against the Exchange Variation.
(sorry about no illustrations, but there were a lot of different positions described, and I assume anyone interested in this thread is familiar enough with the early Ruy not to need them)