You do need some carbs for hypothyroidism. I believe Chris kressler has written about it. But a lower balance would be useful and obvs not the crap ones.
The gluten thing is a myth; it stems from the fact that if you're coeliac you're more likely fo get hypothyroidism. But if you've not got the genetic tendency, it won't make a difference. but gluten is a FODMAP food and its possible that in some it's causing ibs which obviously can help.
I've found middle age/ peri menopause is making it harder definitely. I've also found taking high dose b1 helpful - Isabella wentz has written about it. (also taking some high than usual b12 which is helping brain dog.)
Peri can cause issues with insulin levels which can lead to increasing fat around the middle. Completely avoiding all high sugar foods, eating complex carbs including grains with lots of protein can be really helpful. Vegan forward diet with lots of fibre and some fish and meat. (I love this book and am trying to follow more of it thedoctorskitchen.com/lifestyle/the-doctors-kitchen-podcast-59-how-to-reverse-ageing-part-3-of-4-the-female-ageing-brain-with-dr-lisa-mosconi)
I've also picked up the tip about a good portion of protein before 10 am; eg eggs or chicken.
You also get lower stomach acid as you age and so absorption of the thyroxine can be an issue. Take 30 mins to an hour before food but with a full glass of water if doing so in the morning. I noticed a big difference when it was an hour (but I couldn't sustain it with small kids.)
However everyone is different; what spikes one person's insulin doesn't spike another's. Some people find lots of vegan food too explosive etc.
Years before Michal Mosley got really famous he did an Xmas diet thing and three tips were really useful; eating thick soups; fills you up and keeps you satiated for longer (I imagine especially if proteins in eg lentils) and porridge for breakfast (I add frozen fruit and yoghurt.) also snack freely on almonds as they basically fill you up and pass through.
(And the usual make sure ferritin, b12, vit d and folate are all v good.)