Can you play a single cartridge in both systems at the same time?

It’s the other way around. You can download the same game in all of your consoles and actually play them without media exchange. Something you can’t do with cartridges, because you can only play the game in the console where the cartridge is inserted. So digital games are objectively better for your scenario.

Exactly. Nothing new here. PS3 games were famous for requiring an install from the optical media to the internal drive first, and then also downloading some mandatory major update before running. The role of the physical media was mostly symbolic.