Are there any advantages for a martial artist to learn flips?

Yes there are reasons behind learning these types of things really. Even Mas Oyama believed in they being taught and having some value for traditional martial artist. Purely from an athletic standpoint the athleticism, balance, and coordination that being able to perform acrobatic maneuvers like this also have the additional benefit of making you more physically fit and capable of doing other things including some of those more traditional martial arts type techniques. Also the ability to roll and overcoming your natural fear of falling either forward or backward and rolling forward or backward properly can come in handy at times in self-defense situations. The ability to do flips is just an extension of that ability really.

I have been asked about this before among some of my friends who are fighters in MMA and kickboxing and when I demonstrate some of the ways a roll can be used against them and taking them down more easily or maybe to avoid contact and being pinned up against the ropes or side of the cage and have even seen that tactic applied once or twice before by others in actual MMA fights. A lot of people scoff at these things though because they don't really know or understand the value of being able to do them and a person learning to overcome their fear of going over forwards like when being thrown or backwards when falling or being reaped. That fear can cause a person to stiffen up making the fall much harder on them and maybe knocking the wind out of themselves. Even they just reaching out with their hands to catch themselves can result in things like breaking their wrists, or elbows, or collar bones rather than going with the flow of things and rolling.

No, a jump can serve the same purpose, it's safer and faster, so why even bother with a flip.

No.

Flips are really more for show than they are for practical fighting. If you can do a flip kick, then great, but in a street fight, it's probably not going to help you much.

For self defense? No, there is no self defense application to learning flips.

It's not directly applicable to martial arts, though learning to do them would increase one's overall athletic ability and conditioning. There are probably benefits to proprioception (body awareness) as well.

I've seen some high-level grapplers do flips to counter throws and takedowns, though honestly it's very situation-dependent and there are dozens of other defenses that don't require a high level of athletic ability. If one is doing forms/kata/poomsae for competition, there are certain modern/freestyle divisions where competitors incorporate acrobatic moves, so it might help get a higher score there. But if you made a list of the world's best martial artists, the number who could do such things would be pretty small, so it's not necessary for being a good or great martial artist.

Not really except learning and being able to do it means you have increased your agility, balance, explosiveness and core control.

So it's not the flips themselves but gaining the ability to do them

Yes. You could use them to get out of situations and could use them for a higher kicks or you could use the element of surprise, go between their legs. For self defense, it COULD be helpful, but otherwise, it looks cool... I guess thats about it.