It's a question of attack model. Vaudenay places himself in a model where the attacker is "outside", and trying to work out the password that a human user enters in a Web site (or another similar password-protected protocol). Both client and server systems are honest and truthfully run the TLS protocol; the attacker can inspect and alter packets, but that's all. Chosen Plaintext Attacks are out of scope.
In that model, in case of repeated errors, the human user will consider things too fishy, and will not persist. Moreover, most alterations meant to check an hypothesis on the last byte of a block also mangle the previous bytes in the record, which contain the MAC, so the ultimate observable behaviour, from the attacker's point of view, is unchanged: an (encrypted) alert record is sent by the server, and the connection is aborted (so we are not talking about the difference between a short error message and a long non-error message, but about two error messages, who have the exact same length).
The 2003 paper "fixes" both issues:
It targets the case of an email client that automatically connects to an IMAPS server, every minute, and will try again and again even if connections abort; the errors are also silently ignored.
The difference between failure modes ("bad padding" vs "good padding but bad MAC") can be worked out by the attacker by measuring the server's response time.
Retrospectively, we may note that the problem about the alert message not being known can be avoided when the padding has the size of a full block and the receiving side does not check padding byte contents, as is nominally the case with SSL 3.0. Also some TLS 1.0 implementation did not check padding byte contents.
Around 2010, researchers made some sort of "mental breakthrough" in which they realised that CPA were a valid model in a Web context. This began with padding oracle attacks on ASP.NET, in which the attacker sits on the sending side, and wants to unravel the "viewstate", an encrypted blob that the server adds to a Web page in order to offload state information storage on the client. This is not a TLS attack, but it still demonstrates the "extra powers" awarded to the attacker: the attacker controls the connection and can decide to try again, repeatedly, with no supervision by a human victim; and he gets access to error messages.
Then, in 2011, the BEAST attack demonstrated an attack which was not about padding (though it was still about CBC in TLS), with a CPA model in a Web browser, and targeting a secret value sent automatically by the client (the cookie). From that point, researchers began to really believe that CPA with low success rate per connection were really practical in a Web context.
The same alert message was sent, whether the failure was on the padding or the MAC (also, that error message was not necessarily made available to the attacker).
The MAC was still computed even in case of bad padding, so as to have a constant processing time.
Modern TLS libraries implement really constant-time processing which removes this last timing leak, thus making TLS 1.0 CBC cipher suites safe again (but TLS 1.2 with AEAD suites is still largely preferable).