So is it easy to attach a normal coax to a cheap probe? (to make a separate X1 probe)
And how much more bandwidth can be expected.
Dave simplified (in my opinion oversimplified) this to stay away from transmission line theory. For best high frequency performance, you want to terminate a 50 ohm coax cable in 50 ohms, otherwise you get reflections and ringing. The scope input is 1 Mohm. There is no such thing as 1 Mohm coax cable. Therefore resistive coax is necessary (not just in 10x probes!) to dampen these effects. Attaching a piece of 50 Ohm coax to a 1 Mohm scope input will have a higher bandwidth, but it will also have some nasty ringing in the pass band. The amplitude of signals at certain frequencies might go up by a factor of two! That can introduce some severe distortion. It will also present a huge capacitive load to your circuit under test.
Another important issue is the source impedance. Probes are usually specced when driven from a terminated 50 ohm source, with a 25 ohm source impedance (50 ohm source in parallel with 50 ohm termination). The 100 pF 1X probe will have an impedance of about 160 ohm at 10 MHz, so the voltage at the probe tip will be about 85% of the open circuit voltage due to probe loading. Now connect the same probe to a voltage divider consisting of two 5 kohm resistors across the same 10 MHz sine wave. The source impedance of this point is 2.5 kohm. The signal at the probe tip is now only 6% of the open circuit voltage. This is why 1x probes can be useless even for 1 MHz signals.
or maybe the 9Mohm can be replaced with an 1Mohm?
Only if you also change the rest of the compensation network.
I assume there does not exist a very cheap 1X only probe with like 50Mhz bandwidth.
Not at 1 Mohm.
1X-only probes will use the same resistive coax (it's not just there to optimize the 10X probe) and will have similar ~10 MHz / 100 pF specs. You can get more bandwidth with low-impedance probes, like the 50 ohm coax that Dave suggested. A piece of 50 ohm coax terminated into 50 ohms will have a very high bandwidth, but the input impedance is only 50 ohms.
Edit: this is claimed to have a bandwidth for 60Mhz (no probe just clips)
Terminated into 50 ohms maybe. Terminated into 1 Mohm no.
Some further reading:
Tektronix Oscilloscope Probe Circuits. By far the most detailed work on scope probes. Also contains some information about 1x probes.