Are there drawbacks to installing GFCI outlets in sequence?
That won't work

You're trying to manage sequence of breaker trip, for your convenience of resetting them, and that's tough to do generally.

Especially it isn't going to work with GFCI, because they don't work that way. When there's a ground fault, they all will trip by design. And then, you will have to fight to reset them.

I fail to understand the core problem that you are attempting to solve. They shouldn't be tripping at all. If you have a device that is tripping GFCI, fix it. It's broken.