Are you willing to add bus to the train? On either 46A or 75 routes from DART (Dun Laoghaire station) you have both Monkstown Educate Together NS (multi-denominational) and Kill of the Grange NS (Church of Ireland) schools within 10 minutes on the bus. BOth are busy, as I understand it, but I know Kill of the Grange takes catholics as well, at least, and METNS takes DCs of any and no faiths.
Within Dun Laoghaire itself, there is another school that I don't know much about.
Blackrock has Carysfort NS within about 10 minutes walk from the Dart.
Dalkey village has a good girls school - Loreto - both primary and secondary. It may have more.
There are lots of schools within a few minutes walk from a lot of the southside dart stations. But house prices (and rents) are huge.
We picked where we live for proximity to both buses and dart, walking distance to both. You can now get LEAP tickets, which are like the Oyster card, and buses, trains and Luas trams all operate the same system.
SuperValu and Tesco both do online shopping in the area for delivery. And a wide range of takeaways deliver as well (from burger and chips, pizza, Chinese, Indian, Thai etc).
There is a system called "GoCar" where you can use a car for an hour or 2 at a time, like renting bikes, there are a few cars in the Dun Laoghaire area for that and a good few in Dublin City. And Enterprise and Sixt both have car rental places on Pottery Road (from Dun Laoghaire train station, follow 46A/75 route about 10 minutes up to Bakers Corner pub - Irish landmarks system is often pubs! - and walk about 5 minutes or 3 minutes respectively around the corner to get to the depots). And taxis are relatively plentiful as well if needed.
Open spaces - there is the sea, beaches at Monkstown and the 40 Foot (Glasthule station) for swimming, Killiney beach or Dun Laoghaire Piers for walking, etc. Dun Laoghaire People's Park is small but nice (with playground), while there is a large park at Cabinteely with a good playground and long walk, Killiney Hill also has good playground and range of good walks (and a coffee shop on 1 path), and there are actually lots of playgrounds in the area (Honeypark, Mount Merrion; Dun Laoghaire sea front and Sandymount sea front both have exercise equipment that DCs often use to play with as well).
Cinemas in Stillorgan and Dun Laoghaire. Small theatre in Dun Laoghaire and Dundrum, but also easy access to city centre for major theatres and music venues. Range of extra-curricular activities available locally of all sorts - music, arts, horse riding, dance, drama, scouts and guides, girls brigade (probably boys brigade too), Coder DoJo......all sorts! Plenty of sports clubs, including gaelic football and hurling, soccer, rugby, hockey, tennis, sailing, swimming etc.
We had 1 car when we originally moved, got a 2nd when DD was first born, but dropped it again for a good few years and managed perfectly well without. Only got it back when DH had to spend every weekend travelling away to help DMIL after DFIL died but trains didn't work out and I needed a car here as Scout leader. We still use public transport more than the cars.