Yes because it would open Apple up to someone leaking the details of that deal. After the stand Tim Cook took that would not go down well.

At the start of this Apple-FBI episode, didn't Apple initially ask the FBI to submit their request for unlocking software under seal? Doesn't that imply that Apple was willing to assist? Why else would Apple want to keep the request secret?

Apple said they would assist in unlocking the phone, not the software, under seal. Apple very well could have said they would not modify the software, prompting the FBI to go public and take the fight to the court of public opinion.

Why would Apple want that under seal?

If you had engineers threatening to quit if forced to write "GovtOS", it would make sense that there would be quite a few who wouldn't mind leaking that a behind-the-doors deal was made.

Not foolish, but would Apple really agree to the "we found another way to backdoor the phone" messaging?

What agreement? Apple doesn't nor should they care what the government does with the phone. If the government drops their suit, Apple has prevailed in their argument.

As in - the government is trying to push "Stuff you Apple, we found another way in" argument.

That isn't good PR for Apple.

Whether the government is just lying, in order to save face, remains to be seen - somebody with more knowledge of the technology at hand may be able to comment.