Just to be clear, I think Sol is very good and I use her regularly for things like GO. It was not my intention to look down on her. I put her in the same bucket as Eros, who I also think is very good (even before AW) - there are situations where having Sol or Eros is a real lifesaver, and getting their utility from any other sources is just a real pain. When I de-prioritize Sol for mtix, it's purely based on the observation that I frequently forego Sol for various content, whereas I almost never drop Mike or Tish. Dropping Mike or Tish results in a very noticeable drop in DPS, which limits the content you can clear, how quickly you can clear it and how easy it is to MVP things. Versus dropping Sol, which admittedly makes certain battles very painful sometimes (mostly GO, espeically all 7 himes survive mission.) When it comes to mtix, it's a matter of priority, and my argument is simply that I would rather forego some battles like GO than forego overall DPS.
Also, just to be clear, I am NOT arguing that you should never mtix Sol. If you have Mike, Tish and Lugh already, I would mtix Sol. Also, if you are telling me that you are only ever going to spend money on one mtix only, then I would in fact mtix Sol, but mostly because Sol fits well in rainbow grids and is an excellent hedge against gacha RNG. Mike, Tish and Lugh plays on synergy - they all have potent burst effects and all make the team burst faster directly or indirectly by being fast themselves. If you only have one of them, they just aren't nearly as awesome and Sol starts looking much much more attractive.
Tactics-wise, I fully agree that running Andro to drop Sol is a bad trade, you do not want to do that. When I run without Sol, my strategy often involves letting himes die (usually Tish in my case, sometimes Take if I don't think I can make it until she recharges her swords) and rely on subs (usually Metatron, and sometimes Sol, yes, I DO use her.) This obviously doesn't work for GO if you want all 5 ori, but for any other events, it's just about clearing things quickly, casualties don't matter.
Two other points of consideration that may be causing our differences in opinions:
a. I only have one ascension weapon so far (missed the past light UEs). If you stack more ascension, Sol obviously gets better. For beginners though, they don't have ascension weapons either (the ori shop one isn't exactly great), so it's not like they'll see crazy high heal from Sol as first mtix.
b. I am not at the point where I can carry rag raids yet, so I have not studied their mechanics in-depth to see if Sol is a huge help. Sol is the only hime I'm aware of that can remove enemy buffs from all enemies at once, so if you're telling me that that is sort of required for some rag raids, then I will certainly entertain moving Sol up in the mtix priority. But rag raid is not exactly beginner stuff, so I may still think that you should get Sol after Mike and Tish.
Just tested it myself, replacing a sword with a weapon of similar attack results in a noticeable drop in Mike's ATK. Both the English and JP wiki also say Michael is a sword user (it doesn't have to match their unlock weapon, like Sol is a staff user even though her unlocking weapon is a gun.)
I have not pulled Lugh yet (she will be my next mtix), but my understanding from the JP wiki is that skill 3 is usable every single turn (for 15% hp.) So she can nuke twice every single turn as long as she has the hp to keep up. People with Lugh please confirm (looking at you AutoCrimson.)