I'm gluten-free for a second time (and intending to stay that way) & can say that right now the main benefit I've seen is being able to lose weight. I had my meds adjusted in August and will be going this week to have a redraw to see if my levels are okay on the new dose. I started gluten-free again around the same time I had my meds adjusted. Back in March, 2009 I went gluten-free when I found something on the internet that suggested the same thing you found--a possible link between gluten intolerance and Hashi's. For the first 6 weeks I was miserable, like I was having withdrawl symptoms (shakes, sweats, irritability, insomnia, etc.) but it got better and I felt better. Of course, around the time I felt better I was also walking 5 miles a day 4-5 days a week so I thought that how I felt was more related to the walking. I managed to lose almost 40 lbs from March-August 2009 and attributed it to the walking. At the end of August2009, I gave up the gluten-free diet as I felt that it really wasn't necessary. Around that same time I wasn't able to walk as much. When I started gaining weight and just feeling *blah* I attributed that to not being able to walk as much, not stopping being gluten-free. I've only been gluten-free again for 2 months and I generally feel better + I've lost 15 lbs and not really made any significant change in my lifestyle other than going gluten-free.
I can't have ANY gluten or I feel terrible--bloating, diarrhea, headaches, irritability, etc. + I wake with night sweats, racing heart & have insomnia when I consume gluten. (I thought this was due to my age and "the change" starting but I haven't had this issue in 2 mos except when I have accidentally/unknowingly eaten gluten.)
BTW--What lead me to start again was a consult with the wellness director of my insurance program. He pointed me in the direction of more info on the link between gluten intolerance and Hashi's. When I asked if I need to have periodic monitoring of my TG and TPO antibodies to see if they're going down, he told me no. Even if they do go down, the damage is done to my thyroid and I may very well have to be on meds for the rest of my life. (I was dx'd with Hashi's in May, 2003.) I'll have to continue the bloodwork to monitor thyroid function. The goal of the gluten-free diet is to prevent further damage to the gland, and it may never function fully/correctly again, but being gluten-free won't necessarily "fix" my Hashi's as my gland has been attacked off & on for the last 8+ years.