Tomcats couldn't even shoot AIM-120.

That's a misleading statement.

There is no inherent reason that F-14's could not carry and shoot AIM-120.
It's just that the Navy decided it wasn't worth it.
The F-14 has a long range missile. Why spend money integrating a new
missile on an airframe that's going to go out of service soon?

The changes that would be needed were largely software and flight test.

Yep. Chump change and the aircraft could have had a vastly superior weapon
than Sparrow for a decade. Also better than Phoenix out to AAMRAM's max
range as well. In hindsight it didn't matter ... largely because the
aircraft was underutilized in Desert Storm, its last opportunity to fight
the good fight.

R / John