I personally wouldn't do the gluten thing for that reason.
There's a lot of stuff in the Internet regarding hypothyroidism. I read most stuff with a pinch of salt until I either notice it in myself or find good scientific research. I have never really had an 'issue' with weight gain. I actually loose weight if hypo for too long, as my appetite goes and my metabolism slows. I get very weak. As I said earlier, I notice I look puffy as you get more water retention. I get 'cankles'.
Imo it's not about loosing weight. I think it's really important to be strong / do strength training with thyroid issues as either hypo or hyper causes muscle weakness. Plus cardio to help circulation. This is my theory (when Tsh and t4 and t3 good, though t3 has a very short half life and can vary during the day) . There are a very few people who have got 'conversion' issues. And apparently keeping ferritin over 70, good b12 (and folate apparently) and vit d helps with this - but the other side of this is that deficiencies in these cause similar symptoms and muscle issues, so must be explored. Also pernicious anaemia can co-occur (b12 issues).
Then - strong muscles, more energy, loose weight.
On good levels of thyroxine I used to cycle to work, swim a couple of times a week, do Forrest yoga, capoeira and then took up skipping to improve my stamina. But I noticed if thyroxine was off. There are athletes on thyroxine, Ellie Goulding is and Rachel s club seven (and many more I'm sure!) What I wasn't good at doing was then pacing myself when I had a blip. And you definitely cannot exercise easily when you are unwell. Hoover a room and you need to lie down. But with careful pacing and building you can get there.
My belief is also, if strong, any blips are easier to deal with/ recover from.
I had a blip post baby, my Tsh was 13 by 6 months and I was very weak. Just not quite enough thyroxine. I'd had a section and found it hard to both recover and get back into exercise. I think vit d wasn't great and later found ferritin a tad low. Then more recently the asthma! It's been a long road to recovery as for a while I was on far too much thyroxine (that was hell). It was like chronic fatigue syndrome I think. I couldn't load the dishwasher. I think it weakens the proximal muscles (according to patient. Co.uk) They tested me for Addison's disease. (Not that thank goodness). They decided I have hypermobility syndrome but I have noticed I seem to get weak at my joints when hypo. Now that's recovering I AM very flexible, but strong with it too.
Finally I'm starting to feel well and get strong slowly.
This is just my experience and knowledge after 20 years of the damn thing.
You do need to keep track of your Tsh and how you feel, what you notice. And your levels will need monitoring over the next few months and years, and dose adjusting.
Gone on a bit. Ds asleep in car. Thumb twiddling!!