The FTL for time travel thing is not strictly speaking a requirement..and not possible either, at least with the current laws of the universe as we currently understand them.

One design for a time-travel device is a super dense bit of pipe, set rotating around its longitudinal axis so its circular ends are rotating like a 2d wheel.. and then flying through the middle of it once its fast enough. No light speed needed, just a very large amount of energy required to spin the thing, and some super expensive and hard to source material to make it.

Light speed is analogous to speed of information transmitting.. since the data/info is going to be a radio signal, or microwave burst, or some other packet of EM radiation, all of which propagate at light-speed like visible light itself which constitutes part of the EM spectrum.

Gravity is at the same speed as light - if for some reason our sun vanished one day, the gravitational effect of the vanishing would be felt at the same time the sky went dark. So its not gonna work as an info transmitting device either, even if you had some way to conjure/destroy matter sufficient to generate these pulses.

Quantum entanglement (where you flip a spin on say an electron, and its counterpart also flips its spin in response thereby giving a means of transmitting information) is in theory one way, but to generate a particle-pair that exhibits this property, you then have to separate one from the other and send one of the pair off somewhere.. and its a piece of matter subject to the same constraints as any other piece of matter with regard to speeds etc.

Anyways, thanks for a very off topic sort of question... always nice to refresh yourself on things you rarely revisit.. helps to stay sharp and all that.

And may I also recommend the channel of Isaac Arthur for those with a particularly strong interest in these sort of topics. Its one of the best things on youtube... which for all we know, may not be around for much longer!

https://www.youtube.com/channel/UCZFipeZtQM5CKUjx6grh54g