It is true that Ambien/Lunesta/etc. do not actually induce true sleep but rather a sleep-like state. Your body does not get all the benefits it would get from real sleep, though I agree with you strongly that it is way better & healthier to at least not be awake 23 or 24 hours of the day.
I'm not sure I agree with it, but this does seem common with Headache Clinics based on my own experience with a few of them. They basically want a clean slate to start with -- especially with addicting medications b/c it does take time to get someone weaned off of a med & basically they can't be bothered with that (or so it seems). At the Diamond HA Clinic in Chicago they wanted me to go in-patient for a few weeks but I was teaching at the time so that was just not possible. I went to another HA Clinic (small practice) & they had the same rules, though ironically after all that they ended up putting me on a much stronger opioid that I was on before after reviewing my MRI, conducting a number of tests & trying some of the triptan meds (some of which actually did get rid of my migraines, but I also had issues with nerve pain which triptans don't help with).
My guess it they want to try you on the muscle relaxants b/c they are trying to rule out tension headaches. In any case, the muscle relaxants very likely will make you sleepy so it might be worth a shot. They helped my pain in my shoulder/between shoulder blades, but for me the issue was long-term I seemed to get weaker. But in my case there is a lot going on -- genetic neuro condition, nerve damage, atrophied muscles & muscles that overcompensate b/c of they are neighbors of the atrophied ones, plus I get migraines sometimes [though not nearly as often as I used to]. They got the migraines under control almost immediately with the HA clinic. Then we had the nerve damage which my PM treats. And now I am VERY slowly working on building up those muscles that overcompensate so that they don't cause me pain too (which sometimes can radiate to my neck & head).
It sounds like you maybe have a very complicated case as well. Unfortunately, I've yet to have or meet anyone who has a doctor who can manage all those issues. It takes a team & often they don't agree with each other. When I went, the HA Clinic's view was that some meds -- like triptans -- can actually break a pain cycle. You take one or two doses & the migraine actually goes away. You might get another one, but that one's gone. With muscle relaxants, you take them for a few weeks, maybe longer, and have a chance to rest the overworked muscles & then they (best case) heal and stop causing you pain. Opioids must be taken over & over again and they don't actually stop the pain, they just block the signal.
I'm obviously not opposed to them since I take one, but they are less than ideal. If the opioids are working for you & you are able to live some kind of life, then if I were you I'd find a pain clinic that would work with me & forget the HA clinics and neuros and all that. If the opioids are really not cutting it & you're just miserable beyond reason, (and I've been there -- literally hospitalized for migraines) then maybe it's worth considering going through the misery for 8-12 weeks or whatever it would take to get off of your current meds & into the hospital for 3 weeks at the clinic. Maybe they will find something that the other doctors missed or maybe they know of a new treatment that will work & get you 90% back to your old self. That's always the tricky part, isn't is? To decide whether to stick with what is barely working or to decide to endure even more misery in the hope of finding some kind of cure.
I can't tell you what to do. It's a horrible decision to have to make. But I'm sure you'll make the right decision for you & we will support you either way.