Without a weir door, any time the pump is turned off, any debris that has been collected in the skimmer basket will float out. The door itself does not draw anything in while the pump is operating. If you don't have a weir door on the skimmer, make sure your dump the basket before you turn off the pump.

I have a new pool and my return jet is also much closer to the skimmer than it was on my old pool. Make sure that your return jet eyeball is aimed as far towards the perimeter of the pool away from the skimmer as possible. That will help to create a circular motion of the water and direct floating debris more towards the skimmer. Having said all that, the bottom line is that you are always going to have to do some hand-skimming weir door or not. It is simply not possible for all floating debris to make it into the skimmer.

(In my pool, if I have a lot of floating debris, I aim the eyeball towards the side so that it creates a slight rippling of the water to help move stuff towards the skimmer. However, if there is not much stuff on top of the water, I typically still aim the eyeball towards the side to still get that circular water movement but I aim the jet more downward.)