Can LED Power Supply accept different gauge wires at the same time?
There's nothing wrong with using different wire gauges as long as they can all carry the full current of the power supply. If your power supply could output more current than what each wire is rated for, then a short at the end of a wire would be a fire hazard, and the wires would need fuses, but that's not the case here.
However you're never gonna fit 3 wires in these tiny screw terminals without making a mess. They look really small and close together, the wires are going to be in a bunch and you won't notice the little strand that got out and will cause a shirt. Maybe by joining the wires in a crimp ferrule before putting them in. Whether or not that's up to code is for you to check.
I recommend Wago 221:

But that needs a junction box.
If your strips have accessible terminals, you can always piggyback on the strips by soldering 2 wires to each terminal.

That would be safer than shoving so much wire in this tiny screw terminal because there is much more room do a clean job.
And if the strips have connectors, someone will sell you a T junction.