I use Esbit-style stoves, but with Hexamine fuel that does not leave a residue. I've not had any problem blowing it out when I'm done. I think Esbit is easy to blow out too. Am I missing something here?

For me the main disadvantage is how hard solid fuel is to light. I generally use tindercard or similar to help get it going. Alcohol will ignite with a spark (at least, if it's reasonably warm - it's the vapour that ignites). And for me the main drawback of alcohol is my fear of spilling it while it's aflame.