At the risk of sounding sexist and making an appalling sweeping generalisation, I think it's a common thing for men to not want to give up the car and go sensible - more so when they were previously piss poor and have now got the means and funds to set that right and have their own.
Even still he's gonna have to act like a grown ass man and father and compromise on something that's more practical, sensible, within your budget and is safe for you to be driving around with the kids.
Lot of the more practical, cheaper to run and relatively new, family type cars look good he needn't be forced to endure the shame of owning and driving a bright yellow Fiat Multipla or anything.
My husband has a Mitsubishi Outlander which I didn't expect to like at all initially cos I'm more of a “Buy cars for a couple of hundred quid off a scrapper and drive them until the wheels are falling off and bumper screaming and screeching along the tarmac” type of person but my kids are grown up now so it's just me and the dogs and a lot of shit to traipse around.
I generally use my car like a mobile tack room and cram it with bales, tack, rugs, ropes, buckets and all that kind of stuff so reluctantly agreed to give up my old knackered out Toyota and now have a VW Tuareg which is thirsty as hell, big, bulky and feels heavy and sluggish but I can get anything, everything and then some in it. The space is unreal. Has roof racks and tow bar too so it's easy enough shoving a trailer on the back and practically driving around with your entire family, friends, pets and home in one go.
Interior seats are wipeable and it's easy to just spray, wipe down muddy pawprints and clear out.
It's worth looking at offers available from used dealerships that have only a few hundred on the clock and it can be worth investing in a decent car with full warranty and servicing so you can pay per month via direct debit and not have the worry or hassle of forking out extra for any breakdowns and repairs.
Not just that but the newer the safer they are and will hold up in the event you are unfortunate to have a prang with kids in the car.
Honestly though a lot of the newer electric cars and off-roaders are loads of fun to drive. I robbed my husband's the first time I took it out for what I said was just a drive down the road to get some milk and put it to the test up over the tops at the arse end of farmland and fields. Had a blast, brought it home absolutely drenched in shit and mud. Decided I liked it after all.
Either way up he's gonna have to stop sulking, meet you halfway and just try out a few cars that suit all your needs and be safe rather than stamp his feet and say it's not fair.
Let him narrow down cars within your range, what he would at least be willing to be seen driving and then arrange a test drive. He might come round much quicker and easier.