Problem 1: "Slow down on reentry"... dealing with many times the speed of sound. Turbojet and Turbo fan do not work well at many times the speed of sound, they normally slow down air to slower than speed of sound at combustion stage, so subsonic combustion can keep up with the air. (You are also dealing with vast amounts of energy that would likely destroy the turbines due to physical stresses at supersonic speeds)
Problem 2: Air gets very hot when compressed because of many times the speed of sound, add heat of combustion and materials in turbines can't take the heat.
Normally when talking many times speed of sound, scram jet or ram jets are used, scram jets are very bleeding edge and trying to use them to reverse thrust rather than add thrust would need a redesign and risk more destruction from heat higher than materials can handle.
Buron had engines to speed up rather than slow down, just like a normal commercial jet engine (commercial jets using engines as braking do it only when landing at slow speed for short periods of time). When Buron is down to speeds of a normal jet at end of reentry, the idea was engines could kick in and then Buron flew like a jet to airport to extend options for landing compared to gliding without power.
Normally adding drag... whether parachute, wings, variable wings, changing profile or shape of vehicle, etc makes much more sense than air breathing reverse thrust at mach 10 speeds... You still have issues with overheating, stresses, etc but much less than trying to do combustion and reverse thrust at same time. Once you are down to speeds similar to normal jet planes then normally can land like a jet plane if wanted or regular parachute otherwise (weight not a problem, just need bigger wings or parachute).
Yes in theory you could land like a helicopter rather than like a jet airplane, to allow landing on spot without long runway with an air breathing engine (a helicopter engine could be described as slowing down the helicopter), so far no one has seen need for that.
In case of Musk and friends "reusable rockets"... simpler and probably less weight to just use the rocket engine already there then try to do one of alternatives... and an airbreathing engine with enough thrust to do vertical landing would probably be one of poorer alternatives to gliding down with wings, parachute, etc.
In theory, you could have a very big plane that matches speed and grabs/docks with the reentry vehicle like a hawk grabbing a flying bird as its prey if you could master all the challenges like risk of them colliding. Possible your very big plane could have ram jets or even scram jets and grab the reentry rocket at mach 3+. So far "not worth it" to even seriously think about by guys like Musk making rockets, as too hard and expensive and adds to little advantage.