GFCI test button fails, but tester works?

Since there are millions of homes that don't have GFCI protected outlets in all the required by current code areas, you decide

We made it nearly a hundred years without them. I'd not worry about it.

Personally I would replace it. They cost less then $10.00 and take less then 30 minutes to install. While it is true that most of us have managed to survive without them, a lot of dead people might have lived longer lives if they had used a GFCI. IMHO

I agree.

When in doubt throw it out.

Like the man said they really don't cost that much.

I agree. It's not worth trying to figure out why it's behaving this way. And I'd have to disagree with those that say it's no different than the millions of other non GFCI outlets. This one is in a bathroom, which depending on when the work was done and local codes, may very well be required by code. Second, it's marked as a GFCI. Someone seeing that is more likely to believe it offers GFCI protection when using it. For example, someone going to work outside with an extension cord would be plugging it into a GFCI protected outlet, expecting it to actually be one and work.

It can mean the difference between getting a shock and getting fried to death, but on the other hand you could save $8 to $10 bucks.