I can't comment on all 'brands' of hostel bit generally YHA are great. I've stayed in castles, huge manor houses, restored water mills....hostels opening directly onto beaches, lakes and moorland where I"ve seen wild deer as I stand outside with an early morning cup of tea. You're unlikely to get that with a similarly priced budget hotel. There are generally sociable places to sit in the evening and chat to others about their day and get ideas for places to go. Other children around for yours to play with while you relax nearby.
There are books, board games and often dedicated games rooms with a pool table/giant chess/connect four etc. There are drying rooms for wet towels, walking boots and wetsuits etc. And self catering kitchens so you can save a fortune by bringing your own cereal for the family/making packed lunches or cooking in the evening. If you don't want to cook, most serve reasonably priced food. So the value is in comparing the cost of a hotel in a similar location plus your food costs over 24 hours. Or maybe a travel.lodge further away from the location (usually a souless motorway junction) with petrol and driving time/parking and again buying food out.
Private/family rooms in hotels seem pricey in comparison with dorm beds, but that''s because they more or less multiply the standard bed rate. So a hostel costing £20 per bed (seems like a bargain for a single person) would be £80 for a family in a 4 bed room. And seasonal pricing as well these days. They don't really subsidise families/groups on price as would sffect income and so put costs up for solo travellers too.
So with a family you have to look at each occasion and compare what you need. A two night break somewhere scenic or right in a city centre where hotels are expensive, I would always prefer a family room in a hostel, for all the reasons above. For a one night stop to break a journey or stop over for a wedding, I would go budget hotel.
With baby gear to carry and toddlers to run about after I would veer towards hotel as YHA often have lots of stairs etc and can be harder to carry luggage and keep track of them. From age 4 up, YHA is almost tailor made for their needs.
Experience and memory wise, I have amazing memories of hostels all across the UK but I can barely remember the budget hotels as they were purely for convenience.
Maybe try another when the time and occasion is right?