Hi Shon
I must admit I’m leaning towards your partner’s thinking - of choosing a place to stay that will offer appeal in its own right and be somewhere the kids will look forward to getting back to, running around, cooling off in the pool, or merely being able to grab a favourite toy/book/tablet etc. and find their own ‘chill’. Depending on the time of year, the pool may get used more than you think. Having one in the grounds makes it super-easy as a way to quickly cool off, for kids and adults and it’s quite possible the kids might discover a love of paddling/swimming there.
Staying in the village gives you access to a choice of restaurants, cafes etc. There is little I can recall in La Morra that would appeal to kids (I do recall a tiny little garden/park, but can’t recall exactly where). I see there is now one of the ‘giant benches’ nearby (Panchina Gigante Rossa in a panoramic setting on the edge of La Morra). I’m sure the kids would find that memorable, and maybe another good place for a picnic. The only other thing that might be useful, is if the location has a good popular pedestrianised piazza. I can’t recall anything obvious in La Morra, but if there is, then such places can be an area that people congregate, including their children. I wonder whether the piazza outside Belvedere restaurant might be such a place? Playing with other children is often memorable for kids and they often adapt better than us to language differences - the square in Ravello on the Amalfi coast was one such place.
Will you go hotel / B&B / agriturismo or apartment? FWIW we’re great fans of the latter, preferring to do our own thing, on our own timeline. This might be doubly beneficial with kids so that you can adapt to their eating schedule. That doesn’t mean you won’t eat out, but you might choose to eat out at lunchtime, and then have selection of foods in the apartment, topped up every day or two, that can also double as a picnic to be had on a vineyard walk (these are open to all, and a joy to walk). It also allows you to kick back in the evening to drink one of the bottles acquired earlier in the day / holiday. Food shopping can be at local alimentari / specialists, or alternatively there are loads of big 'out of town) shops including hypermarkets (e.g. Bennet) lining the main road between Brà and Alba (ss231).
There is a place fractionally further out (maybe 10-15 mins drive) in Santa Vittoria d’Alba, that we stayed in with a pool in some pleasant gardens (see my avatar!) with a patio / bbq area, plus a resident cat or two. Very much recommended and I think gives that space and calmness that might be necessary. The place is called Val di Spinso.
Are any of the other villages better options? I’m not sure they are. Perhaps Treiso near Barbaresco has that nice quiet feel (and a great selection of places to eat).
Italians are typically wonderful with kids, but nonetheless after they’ve seen the first barrel store and bottling line, interest may wane quickly, and ditto while the grown-ups do their ‘tasting’ thing. Thus it may pay to limit the winery visits to max 1-2 per day (expect each to take ~ 90 mins give or take a half hour). Ideally not every day either. If you can find any who have similarly aged kids then that might be well worth leaning towards them.
Other stuff to do in the area? Alba certainly worth a visit, with lots of historic buildings. The local tourist office (langheroero.it) are brilliant, and there is stuff like balloon rides that might make for some big excitement / a memorable experience for the kids (and adults). Worth a detour as well to the Cappella delle Brunate (a very bright and colourful renovation done by two Americans, with payment in cases of Barolo). There are castles, though how much that appeals to a 3 and 6 year old I’m not sure.
