They don't always respawn right away, but they do exist, I can tell you that. On my new private server, that's how I got my first wyvern egg, was a low level 50 ice and I pumped it full of stam and got a 150 fire, and a 175 fire that I'm raising now (along with an 85 because I need a male to breed them.)
They don't stick around the nest areas, so an egg doesn't always spawn. It's best to hunt them down and kill them and hope for a fresh spawn. If you really want one that you see wild to lay an egg and you're _really_ lucky, you can kite one to one of the nesting areas and distract it with a mammoth or something, leave render, and it will have laid an egg. I suppose you could trap one by a nest as well, which would be easier, but I never tried it so I can't guarantee it wouldn't despawn when you left render.
Another poster on here mentioned he'd been able to kite them all the way to the Wyvern cave and get them to follow inside, and they would lay eggs in the wyvern cave.
Unfortunately, information on this game is a mess because of the number of changes WC has made over the years, and there is so much old information still on the internet and much of the wikis haven't been updated. Originally, they even spawned in the Northwest snow range above the ice cave (along with one of the treasure chests, now there is only one chest left, that I know about), but they were relocated a long time ago.
I saw a video showing 4 coordinates where Ice Wyvern eggs spawn, but even during the video he couldn't show the exact nest location for at least one of them. Of the 5 coordinates, I've only recently seen nests appear at spots 1 & 5 (C&D on the map). In the past I've seen them at 3 (B). The other two spots I can't even confirm if they are good or not: https://www.google.com/search?client=firefox-b-1-d&q=ark+ice+wyvern+coordinates
MAP
Wyverns on Ark have ever been a mess because they're one of those creatures like Raptors that just get copied and pasted by Wild Card, which is why during Fear Evolved 2 years ago, the Ice Wyverns were replaced with a massive overspawn of Zombie Wyverns, and why the Ice Wyverns on Valguero couldn't be cryopodded and the wild ones were completely invisible for a while when Valguero came out.
I've also seen people build little metal huts near the egg spawns to fast travel and steal and egg, but if they build too close to the nest, they could ruin the spawn completely.