Welcome to the forum

There is some confusion over what does what in an automatic transmission
A band is a brake, it stops parts inside from spinning, it is not like a fan belt that turns things.
Clutches work like manual trans clutch, engage and disengage, a clutch will disengage part of the transmission and then the band will hold from moving.

Automatics run on Pressure, pressure holds and releases the bands(brakes) and clutches
To engage Reverse requires the highest pressure, 150+psi, this is why slow or no reverse is a common issue on automatics since they were first used.
Internal seals and gaskets can start to leak, and some pressure is lost, when it gets low enough then Reverse starts to be slow to engage.
The pressure is supplied by the Front Pump, Front pump is powered directly by the torque converter's outside case which is bolted to the engine's crank shaft.
So when engine is running Front pump is on, raising engine RPMs also raises Front pump speed so increases pressure in transmission, this is why REVing the engine can get Reverse to engage when it wouldn't at idle.

On the outside of all automatics there are test ports so internal pressure can be tested.
You need a 300psi gauge to be safe
Most transmission shops will test pressures for a fee

The valve body in the transmission is what routes the pressure to the clutches and bands, and this is the most common area for leaks, loss of pressure.
Valve body can be changed just by removing the pan, but you would probably want to test pressure FIRST, because there are internal parts that can break/wear out.
So if pressure is good and no Reverse then trans will need to come out
If pressure is low then valve body repair might fix it

Solenoids can not prevent a transmission from going into gear, they just allow for smoother shifts
DTR sensor is just to let PCM(transmission computer control) what gear driver has selected, it also can not prevent "in gear"