I imagine it is more of an exponential decay mixed with poisson since strikes were far more common back in the day. Also, I'd guess an exponential decay in the expected size of impactors over time as they've been smashing themselves into pieces.

Likely it isn't, because the Solar system today and 3Bln years ago are two very different systems.

Have the data actually been fit a Poisson distribution? Or is this is just a guess assuming constant rate and independence?

No natural phenomena ever exactly fits any probability distribution.

except the emission spectra from atoms :)

Right but I'm saying do we have data showing it's even close? (Genuinely asking, I have no idea.)