I'm going to make some assumptions.
1. You said "home" so space and weight is not a big consideration.
2. You said "EMT" so I'm not going to address ouchie first aid concerns.
3. You're willing to get some training. Maybe not a lot, but some.
Based on those assumptions, here's my suggestions.
Gear:
1. A few Israeli Battle Dressings, Olaes Modular Bandages, or similar.
2. Some kind of tourniquet. (Perhaps one you can apply solo...)
3. Some kind of occlusive dressings. (Olaes includes one.)
4. Extra gauze. Rolls are good for packing but pads are useful too.
5. Duct tape. Yes, I'm serious.
Training:
1. CPR. If nothing else, take a basic CPR class as a family.
There's lots of other gear you can buy and many training opportunities, but for a family on a limited budget (money and time) those are my suggestions. There's youtube videos to show you how to use an IBD or Olaes. If you want to get QuikClot that's fine. If you want to buy an Asherman Chest Seal or Bolin Chest Seal, go for it. If you know what to do with a NPA or ARS then add them too, but then you probably don't need my advice.
