***"Is it possible the vent hole in the old cap is not venting enough air to allow the gas to flow to the carb?"***
I suppose it would be appropriate to say "Well DUH!"
because your plight, and the entire description/scenario is probably the most often reported operation related complaint on this forum.
Clean the vent holes with a stiff wire (think paper clip) and blow out with compressed air if you have it. Probe the hole well, from both outside the cap and inside (if the inside hole is accessible).
Then report back your results.

You didn't mention what ENGINE you have on your machine. I believe many of the RX75's have Kawasaki engines, but I don't know if they all do or not. Here's one thing to consider, particularly if you happen to have a Kohler engine: Kohler fuel filters come in two configurations (although they look pretty much alike) - for engines that have a fuel pump, and for engines that are gravity feed. In the case of Kohlers, if you put a fuel-pump-type filter on a gravity feed engine, the engine will run for a little while, then die. This is because gravity cannot feed enough fuel through a fuel-pump type filter to keep things running. I have no idea about YOUR engine, but perhaps it's possible you put the wrong type of fuel filter on. Your problem is an ENGINE problem, not specific to your MOWER. THIS IS WHY YOU NEED TO IDENTIFY THE SPECIFIC ENGINE the problem relates to.

Lawn mower and tractor manufacturers, generally, do NOT manufacture the engines. If an engine won't start or run, that's an ENGINE problem, and forum members need to know exactly what ENGINE you're having a problem with. Matter of fact, most mower manufacturers don't actually MAKE many (if not most) of the parts (think engines, transmissions, belts, tires, deck spindles, wiring, batteries, filters, etc.)

I suggest that you verify that the new fuel filter you installed is the correct one.

It would be a Kawasaki. Hopefully, you had verified the valve is also clean. If the cap seems to be venting, it is possible that you are chasing a problem that is not fuel related after all. there isn't much "head" on these if I remember correctly and it may not ever be able to keep the filter full of fuel. The IPL shows no pump on your model. Is this correct?

I had this problem when I first got my RX-75. The previous owner kept it in a barn. As it turned out the air filter was completely caked over in dirt. Easy remedy. The foam around the outside will slide off (be gentle) and use just a little dish soap under the sink and use your fingers. Squeeze all the water out and let it dry. The other part you can tap on a hard surface to get most of the dirt out and you should be all set there.

I also had a similar problem when I changed a back tire. The fuel cutout valve got bumped. Just the slightest turn was all it took to give me a headache and most of my afternoon shot. Good luck.