I have a huge Hardigg case in my barn (100+F to -10F)
would it hurt to store my gear (packs, cammies, chest rigs, etc in there long term? I know short term a year or two is not bad but not sure if it would affect the coatings on the back of most fabrics.

I'm running out of space in our small house and I have a crap ton of older packs and web gear that I don't use but don't want to get rid off either.

I know ammo is a no go in the barn but not sure on tactical gear.

It shouldn't. Biggest factor is moisture. If you can store them dry in a drysack with moisture absorbing materials, like some silica packets, you'll be fine. Anything that can't handle that temperature range is garbage.

Coatings tend to break down over time with heat especially rubberized or acrylic fabric coatings.

This. This will turn all your DWR treated cordura kit into kit that feels like you spilled soda all over it. It takes years though

This is what happened to my original sea bag from the 90’s about 15 years ago.

Not risking it. I’ll just store my camo netting  maybe, extra cammies

I store a lot of my hunting clothing/gear in large (150-165 qt.) Igloo ice chests, which can be picked up pretty cheaply at Costco, Sam's Club, etc. They work very well for temperature regulation, as you'd expect.