TYPO3 Core

If you end up with a database connection error, the reason for the problem might be somewhere else:
Either the server has too limited resources available, or the database connections are not closed correctly by PHP.
So you should try to investigate this problem a little more.

Additionally, the "fail-safe" mechanism you have mentioned is possible already by using the [FE][pageCacheToExternalFiles] option in TYPO3s Install Tool.

Closing.

I am using pageCacheToExternalFiles=TRUE.

When I stop mysql server, Typo3 displays: "Database Error; The current username, password or host was not accepted when the connection to the database was attempted to be established" instead of real page.

Yes I know. Please see my last comment: TYPO3 has a problem connecting to the database.
You are using the file-cache feature but still get the error because there are still some parts on your website which cannot be cached. They will require database connections for every request, independent of your cache settings.
To dig more into this problem, please enable database logging and check what queries are being executed...

Besides of this I would like to ask you to not reopen bugs again and again. The bugtracker is probably not the right place for discussions like this or in #0002673. So if you disagree with the resolution change of one of the developers, please do not change it again unless you have assured that the problem really exists. I suggest you start a discussion on typo3-dev for things like this....

One more note about this: Parts like the search engine which are purely dynamic can never be cached and would never work without a database access. In this case, the fail-safe mechanism idea is not a perfect solution anyway...

Since it cannot connect to database, it has to display exported HTML, no matter what.

Stucki said it all.