IIRC you can get a 4007 to around 50MHz, but only with a 15V rail and with a small amplitude output. The single chip packages in ALS should get past the 100MHz mark, though it likely will only drive a very low capacitance load ( not a scope probe unless it is sub 1pF) then. You might need to use another as a buffer, though the signal out will most likely not look anything like a nice square digital signal. LVDS differential drivers should get past 400MHz though for some, as they are designed for high speed.
Modulating the supply will change both frequency and amplitude, and it is a toss up as to which will be greater. the crystal will do AM because it has a very high Q and does not change frequency much with drive, but the LC will vary a lot, as gate capacitance makes a change.
Try it using a signal generator with an output offset capacity, and drive it using 1V on a supply of 5V. then see if you can see FM on the output on the scope, or use a spectrum analyser.