OP you are not alone and honestly it's exhausting sometimes always having to think about having enough to pay the next bill. People's finances are so different. I could have written your post though I'm at the other end of my 30's and am very fortunate to be on the property ladder, something that happened by sheer luck around age 33 I believe. I don't think home ownership is the be all and end all, but I am very thankful I own and for me because there isn't a lot of financial security in other ways, I feel I especially appreciate knowing my home will one day be my own. I have affluent friends who only rent because they want to travel and be free to move and do as they please. But they also have fat bank accounts and could buy as needed.
Some of my most comfortable friends don't have as much as some of my friends who feel in a financial crisis. I have a dear friend whose family is on 25K/year with two children. They have a lovely two up, two down with a kitchen extension. They live a simple and good life. Like me they felt better when they could own, and moved to do so.
I do think it's very hard to compare having kids young and being off the career ladder with people who worked and saved two incomes for 10+ years before having dc and therefore had a decent down payment, owned a home, had some savings. And likely sometimes you will feel you are the only one for whom those things aren't true, but you aren't. Plenty of people find themselves in tougher spots due to having kids earlier, divorce, death of their partner, bad health, bad luck and many other reasons.
I would encourage you to set a small but achievable goal. It could be something like saving 1000k. Something that helps you feel more secure but is realistic for your situation. I also think there are likely ways to top up your income a bit now. Could you clean? Help baby-sit for friends or neighbours? Deliver papers? Deliver pizzas? There's often something, even if family have to help with your children if your parter is away for work or has long hours.
DC can have a lovely life without lots of money, but I do think parents are less stressed generally when they know there is money (extra) in the bank and that helps with everyone feeling on top of things, calm and generally better all round.
Good luck to your family.