It looks to me that all that is required to replicate this after getting a new chain and hook is to cut the bottom link of the new chain and use pliers to twist the broken link to the rest of the chain after first inserting it through the eye hook. Is this technique correct?
Don't cut the chain. Either use a repair link/coupling, or slip it on the eye.
If you spread the eye, it won't come back tight, as it will spring away from the point of contact. The secret to keeping it tight is to twist the eye open - use a lever/rod/wrench to move the free end away from the shank (sideways), then slip on the chain, then twist the free end back into place.
I suppose there is more than one way to do this, but I'd be strongly inclined to spread the eye-bolt slightly, fit in the twist-link chain, and the squeeze the eye bolt head closed again.
To do the spreading, I would use whatever comes to hand to act as a wedge. Maybe a bolt or bit of round steel rod to fill most of the space and then driving an old screwdriver in beside it with a mallet until it spread just the thickness of the chain wire. A wedge or cone of wood might do the trick too.
Another approach that's perhaps a bit more violent is to use a strong rod or tool handle passed through the eye and lever the end of the eye off to the side, kind of making it a bit of a spiral instead of a circle. You'll need to move it further, but it might be easier to find your tooling. Just stick the rod in the other way and bend it back to close things up. Only bend it as far as necessary to pass the chain to avoid fatiguing the metal
Avoid scratching the galvanized coating if you can as that would create a point for rust to start and perhaps shorten the cosmetic life of the assembly.
For closing it up, a pair of channel-lock pliers perhaps, with a scrap of wood on each side as bearing. You may also be able to gently beat it closed with a rubber, plastic, wood, or rawhide mallet.
I think this definitely qualifies in the category of "a million ways" to do it, but they would all be variations on the spread, slip, and close theme :)