First-off, congrats on becoming a dad!

The best job a guy could have.
Re: a .22 "that just gets the job done"~the 3 rifles you've listed wouldn't be on my radar personally, and I've owned a 10/22 (bought new) as well as a Henry pump...I've shot a couple of Henry levers, and believe that the CZ452s (predecessors to the 455s) were better-built guns. However, opinions "are like a_ _ _ _ _ _ _" they say~everybody has one.

Your money, your choice. All 3 have huge followings so my opinion is just that~an opinion.
As for what to get~clearly, the best value can be had if you're open to the idea of "used" of course. Try as I might, I personally can't warm-up to Cooeys that much...and I've owned at least 3 different models..maybe 4. I mean, simple..robust things, just not my cup of tea. As far as I'm concerned, their charm is about the most endearing thing about them. I'm also not a fan of semis for a host of reasons, and I've had a bunch of them. Down to just one, a Marlin 795 that only sees the light of day a few times a year.
Budget rifles~there are several options, but to many...Savage would be pretty hard to beat. Mark II series, available in heavy or standard weight barrels, blued or stainless steel. Spare magazines are affordable and readily available. The heavy-barrel versions would be a poor choice unless you shoot prone, or know you'll always have a rest (tree branch?) close by. Offhand shots with them would be an uphill battle. Marlin bolt guns are also tough to beat for the price and personally...I prefer them a bit over the Savages. Just a little less common to find too.
If your budget allowed for a CZ ($500-$600) then a 452 American would be awfully tough to beat. It's a standard-weight barrel, better for off hand shots/long hikes. The model is not equipped with iron sights, so scopes are a must. If you were set on using iron sights, then a 452 LUX (or BRNO 2E, it's predecessor) would be a superb pick. You could scope one of these down the road, but the shape of the stock is better-suited for use with it's iron sights.
If you have any buddies with 22s...see if they'll let you try them. Often times, it's impossible to explain why one is better than the other...but in trying them, you can sometimes get sense of what you're getting for the extra $. I don't know what the OPs sensibilities are...but some people are well-served by a $100 Cooey their hole lives and never want anything else. Other people have 20 or more 22s and are still trying to find the "perfect" gun. I think for hunting~a simple bolt-action repeater is the best combination of accuracy, ease of use, dependability, easy to clean, etc.