A word of caution regarding any description of power amps would be that any opinion, comment, or marketing info that states an amp has a "clean" response has never meant that the tonal response is flat... and your AFIII/FM9/FM3 is expecting a flat response in a separate power amp.
So, if we are looking for simplicity of operation and/or authentic tones, I see no reason why, when we have the best amp modeler on the market to then go use a tube power amp for monitoring. You may like the tone, or prefer it to SS, but it will not be tonally authentic, nor a direct relation to the tone you send to FOH... so why add confusion?
There are a number of well known SS amps that have been used for years (such as the Matrix et al) however, designs and technology have moved on, and frankly with the $$/W that a class D design offers, it is todays most logical choice for solid state power amps.
With ruler flat frequency response, low cost, large power outputs, small size, lightweight and high efficiencies there is nothing to dislike.
There are any number of options from a straight power amp, or amp amp with some Eq, but you can also use a number of class D bass head via FX return or aux in, or you can even build your own for low $$ if you have basic soldering skills.