Does it ever make sense to use a 50Ω input with 10X scope probe?
A high impednace passive scope probe treats the cable as a capacitor rather than treating it as a transmission line. The compensation capacitance in the probe balances (with the appropriate scale factor) the capacitance of the cable and the capacitance of the scope input.

The cable on a high quality high impedance scope probe is special, it's not normal 50 ohm coax. The special cable along with the relatively short lengths of probe leads means they can get away with treating it as a capacitor at up to 100 MHz or so, much beyond that and traditional high impedance passive scope probes don't work too well.

Using a 10x probe designed for a 1 megohm scope input on a 50 ohm scope input doesn't make much sense.

The alternative to high impedance scope probing is to run a 50 ohm line to the scope and run the scope in 50 ohm mode (or use an inline terminator if your scope is too cheap to have a 50 ohm option). Compensation capacitors are no longer needed. 

If 50 ohms is too low for your application then you can add a series resistor at the point of probing. For example adding a 450 ohm series resistor would give you an x10 probe with a 500 ohm input impedance. Adding a 4950 ohm series resistor would give you an x100 probe with a 5 kilohm input impedance.

The great thing about low impedance probing is you don't need compensation capacitance and the line back to the scope is a regular 50 ohm line. So it's much easier to integrate low impedance probing into your design than it is to integrate high impedance probing.