That's a great way to make a fighter jet even more fantastically expensive while being less effective.
One obvious problem is latency. 1000 miles is about a 10 millisecond round-trip delay alone even before you add the time it takes to capture and trasmit the wide-angle high-definition images required to replicate what a pilot could see from the cockpit.
Another obvious problem is electronic warfare. A fighter jet with a pilot is difficult to impossible to hack. A remote-controlled fighter jet is possible to hack. You can make this difficult, but keep in mind that you are facing state-level actors, and if they figure out how to hack one jet, they can hack all your jets. That's billions of dollars of jets that are suddenly useless to you, and useful to your enemy.
Hacking aside, jamming is also an issue. An adversary does not have to take over your jet- if they can disrupt communications they can crash your jets.
and, for that matter, recon and bomber aircraft
There are remote-controlled aircraft with these missions. "Fighter jet" is a uniquely bad mission for drones.
