I have a chamber adapter .308 (7.62x51) to 7.62x39. It's been here for ages, and is still untested. This is slightly different than OP's, as the adapter is supposed to stay in the barrel. (held in place with locktite so it can be removed later)
I have also seen/used various inserts for shotguns, and like your style, they are cycled with the action, and the sub cartridge removed manually after cycling. They do work, but there isn't much rifled barrel. The style OP shows might take advantage of the parent rifle's barrel, which is nice.
Awesome method to keep costs down and practice with less recoil, or noise. Great for smaller, or new shooters.
When I first started inquiring about chamber adapters, it was made clear to me by the manufacturer (Ace in Alaska) that .303 to 7.62x39 was for blanks only. Using real 7.62x39 would degrade the thin adapter.
As I mentioned above - I'd be leery about using an adapter designed to fit bullets through a gun that are too big for the barrel bore diameter.
Your x39 bullets are actually supposed to be 0.312 (thought it was supposed to be more than 0.310 or 0.311). The bore on your 308 Winchester is 0.308 or so (some are a few ten thou bigger or smaller). That's a fairly significant difference....
It certainly might fire, but depending on the type of ammo you're using, you're very liable to get chamber pressures that are too high.
Even with cast lead bullets, you generally don't want to go 3 thou over bore diameter. 1-2 thou over is about the most that most people go (I've seen guys with bigger bore 45-70's using 0.461 in their 0.4585 bore diameter guns, and that's about the max I've seen people say they use).
But with jacketed bullets, 3 thou is a lot. If the jacket is really soft, and the core is lead, then sure it'll swage down, but your accuracy will likely be horrid (not to mention the massive jump to the rifling you'll see). But swaging down that bullet is going to take force, so the chamber pressure will certainly spike. How high it spikes, and if it's dangerous, is up to you - I personally wouldn't ever do it though.
With steel/bimetal jackets, I would argue that it could be VERY dangerous to use these in the too-small bore. Add in the steel core aspect that the bullet might contain, and you're looking at a bullet that won't compress very well at all. I wouldn't be surprised to hear about blowing up a gun using this kind of 7.62x39mm ammo in a 308 winchester bore.
In fact, I'm surprised someone actually sells an adapter like this - that's kind of insane.
If you really want to shoot cheaply, with cheap ammo, better to just buy an SKS or similar gun that runs cheap ammo, and have fun with that. The cost of the adapter likely isn't small, and then you're running ammo in a gun not designed for it, that isn't exactly quick or easy to load and unload and reload.
Each to his own, but you wouldn't catch me running it through my guns.
