Ironically I thought the same as you when we bought in 2004. Friends who'd got on the ladder 2-3 years ahead of us had paid a tiny percentage of what we then had to pay for our first home, and it felt like we had a mountain to climb in comparison. But really, you just dig in where you can and as the years tick by, it gets easier. There will always be people better off than you, and less well off. Don't compare yourself to friend X who is lucky enough to "only need" a 2 bed flat. If you want to be home-owners, accept it'll involve compromise, or stick with renting.
We felt we "needed" our first house to be a 3 bed too, because house prices had risen so much I was terrified of negative equity. We found literally the cheapest 3 bed in Woking and bought it. It had dozens of compromises. The location wasn't the best, the smallest room was 6' X 7' and one wall was made of dodgy 60s glass. You cut your cloth to what you can afford and over time you work back to where you'd rather live. For us we were massively priced out of anywhere near the station, so we sucked up bus journeys and long walks for several years. We could have bought a 2 bed closer in, which in retrospect might have been more sensible, but the 3rd bedroom won out for us.
It sounds to me like you're not prepared to make the compromises you'd need to, to buy a house in your budget. That's fine, you can stick with renting, but I don't think you're in an unusual position. Compromising (a lot) is part and parcel of buying their first home for an awful lot of people, not just today but in the past too. Buying in their first, second or even third choice location, or going straight into a 3 bed house, is a pipe dream for a lot of FTBs. It's a bit more complicated as you have children already but it's still doable, you just have to compromise differently. Eg maybe look further afield in area, or more rurally, or look for a nice 2 bed flat with flexible layout where you could split one of the bedrooms or living room into 2 bedrooms. If you get your head round the fact that there WILL be compromises then you can make it work. But it's your choice. A cheaper BTL flat elsewhere is a plan B if it still works taxwise.