It's been done: https://en.wikipedia.org/wiki/JATO
The primary requirement for a carrier jet is two engines, decent thrust, and enough body rigidity to take landings. The F-22 has all that. The MIG-29 had less, and had a carrier version. Same with the SU-27.
JATO is for takeoff, thrust in the aft direction is not the same as trying to rocket assist a landing. Completely different. It's also extremely hard on airframes.
The idea is not feasible.
Also your list of "carrier aircraft requirements" is incorrect, the JSF is a single engine aircraft, and landing gear is a major requirement of a carrier aircraft. The F22 is not a naval aircraft, was not designed to be, and never will be.
He is correct about carrier aircraft requirements. The navy was forced to accept the JSF.
As produced, the F-22 is obviously not a naval aircraft. The tail hook is single-use, the landing gear is not reinforced, and the more modern stuff for carrier approach is probably not installed. All of that would be easy to change, and in fact a carrier version was proposed.
The Navy spearheaded the JSF for both its CVN fleet and for the USMC. The JSF is not the first single engine Naval aircraft, and its engine is reliable enough that it's not an issue. People look at the tomcat and hornet and make up requirements that have never existed.
There is no such thing as "easy to change", you would have to redesign the aircraft from the ground up. You don't "navalize" an air force asset, you have the air force use Naval aircraft if you want dual use. The F22 is not and never will be a carrier aircraft.
Oh come on. You're just dismissing it, seemingly with the assumption that we'd have manned aircraft dropping into tight aircraft parking spaces with all the grace of the very first SpaceX landing attempt.
Obviously there is no reason to bother with vertical landings on a fully functional full-sized US aircraft carrier. This would be for other ships, clearings in jungles, and cleared-out parking lots.
You'd descend toward an area that has been cleared of debris and personal. It's not more absurd than flying toward a ship at 135 kts and expecting to grab a cable without crashing into things and people on deck. Compared to what we do on a CATOBAR ship, rocket-enabled descent is really tame and safe.
Vertical landing with the F-35 isn't exactly safe. This is the standard for comparison. Rockets can respond faster. This allows better stability and faster shut-down.
We have aircraft designed to do stovl and do it correctly.
Vertical landing with the F35 is exactly safe. It's the definition of safe, it's been done thousands of times with zero incidents. It was done hundreds of thousands of times in the Harrier before, and they applied the safety lessons learned from the Harrier to the JSF. How many rockets has spacex lost already?