The Sheraton TST has just been updated, and I'd say that it would historically have been more kid friendly as an area as it isn't a bar area. However, right next to the Renaissance now is an excellent Harbourside park with several miles of greenery and little things for kids to do along the waterfront. I take my kids here regularly, plus right after the conference centre you have Tamar Park, and it's a lovely walk by the harbour to the star ferry to TST (in central, or get the one very near the hotel in Wan Chai). Since examples here:
https://www.hfc.org.hk/en/hss/water-...cinct-wan-chai
https://www.hfc.org.hk/en/hss/harbourchill-wan-chai
https://www.hfc.org.hk/en/hss/the-re...t-causeway-bay
I'd highlight that exhibition station opening right next to the Renaissance has also made the MTR far closer, though the light blue line is a little less handy than the red line.

But, neither location is going to be great if the main focus is Disney, as you'll end up with fairly long transfers to get to the Disney line. Instead I'd normally recommend the W or Ritz Carlton above Kowloon station, or even the Sheraton/Four Points near Tung Chung (fairly cheap cab ride away). Or just accept you won't be that close to Disney and consider the Ocean Park Marriott (note classed as a resort, so no lounge but a good pool) that gives you that theme park, though even more transfers to get to Disneyland.

On the lounge access concept, with one young kid nobody ever had an issue for me, so if there will be 2 adults and 2 young kids I suspect no significant challenge (a third adult would likely be). But as mentioned by others, the Renaissance is definitely hard to get into on a capacity level.