For just tuning the FM coil, I'd use a high-power opamp by itself as the Micro Lambda appnote suggests.
Tuning the main coil is where things get interesting. ADA4870 could handle that in many cases, but you absolutely do need feedback that represents the current through the reference resistor. The Micro-Lambda app note skips that in the FM coil driver because they probably assume it will be operated in a PLL (will yours?)
In this circuit, the lag introduced by the tuning coil needs to be compensated in the - feedback loop. Look at the schematic of some of the old HP sweeper plugins like the 86222A, and you'll see multiple series RCs for that purpose. You don't necessarily need to do the math, just drive it with a square wave and pick some RC components that minimize the ringing.
There should be a small resistance at the MOSFET gate as a parasitic suppressor, and there should be a larger resistance in the return line to the opamp's + input to keep from exposing it to high flyback current.