1) Saiyans engaged in a period of expansionism across the galaxy after Freeza's fall.
2) That they were later completely scattered by King Cold's retaliation.
Then it's not nearly as random and without reason as you seem to think. When Raditz arrived on Earth, he knew that Kakarot had been sent there. So it's not unreasonable that, in their period of dominance, more Saiyans would have heard tell of where their own kind had been dispersed to. Thus, they'd be up for checking it out. So, by all means, the PLOT of the RP makes it very reasonable for some Saiyans to arrive on Earth.
Like I said, there were about four Saiyans that came in when this was originally run here, and only one of them was even remotely a gamebreaker. It's not the race that becomes the issue, but the people who play them. I'm more than savvy when it comes to character growth, and I can assure you that I wouldn't be out for a power quest.
But in the end, if it's the will of the majority, I will refrain, at least for now, from using a Saiyan character.