"Sorry could you repeat that? Someone in the next town over clicked their pen and resulting change in gravity due to the ink tube offset by 1cm drowned out your signal."
Yeah, now try to measure a very weak accelerational signal with a reasonable size (even at high frequencies)
Remember how big and complicated LIGO and VIRGO are and that's to detect the loudest signals of that nature in the universe (at a very narrow frequency range)
It’s not dumb at all and perhaps some incredibly advanced civilisation would do exactly that. However space time is exceedingly stiff, the energies required to create measurable ripples are enormous. Targeted neutrino based communication would probably make more sense though.
It's not very practical. We've been able to detect gravitational waves from black holes (or maybe neutron stars? Something like that anyways...) falling into each other, so in theory we could send a message across the galaxy using non-electromagnetic means if we could deliberately cause a similar event.
I'm not sure why we'd want to even if we could, though I guess "hey look at this, I can push a black hole into another one" would likely be regarded as a rather meaningful kind of message on its own to anyone with the technology to hear it.
Gravitons [0] are the only particles that would be able to travel in higher dimensions, so yes if they do exist along with higher dimensions that you find yourself in need of communicating with then gravity would be pretty useful [1]. You do need to move around a planet or at least a moon to generate the signals though.
