I don't think there's a magic formula or trick you're missing, IMO to make competitive riding + job + being a parent to small children work you need help with the children, the horse or both. TBH I'm impressed you're managing to keep on top of just looking after the horse on DIY without your DH around unless you only work really part-time and have the kids in full-time childcare?
The pre-school/toddler age is the most difficult to have around horses IMO, smaller than that and they could be carried around in a sling or parked in a buggy whilst you do at least some jobs, older they can be trusted to sit somewhere warm/sheltered with a screen or a book without having your eyes on them 100% of the time for a little while, but at the pre-school age, no chance. But it will get easier as they get older, for sure.
But it isn't a binary 'ride competitively and win loads of rosettes' or 'don't ride at all and just enjoy them as animals' thing, how many hours can you spare for the horse a week? I'd prioritize what gives you most enjoyment within your time budget, whether that's hacking or competing or socialising or just spending time around the horse. Personally I do like to compete but turnout and looking good is the least important to me so I'll happily turn up at the yard at the last possible minute, bung pony on the box and roll off to a clinic or local comp, he's clean enough for the tack to be comfortable and we don't trail straw/poo/mud all over the lovely fancy venue or anything but spending hours bathing and plaiting and polishing just for an unaffiliated SJ or a fun ride or clinic ain't for me I'm afraid (helps if your horse is dark coloured, of course mine is grey so my slovenly ways are out there for all to see). But I know some people find all that very therapeutic so if that's you then fair, but if that means an outing turns from a 2-3 hour thing to taking all day than you might need to do it less often. Nor am I as competitive as I was when younger, I have a less expensive horse, less prep time and not as much professional help as others so not winning doesn't bother me (although it's nice to see scores/results improve with practice). Getting the horse (and yourself!) fit enough to do occasional fun rides or unaffiliated hunter trials or XC schooling where you can set your own pace and pull up when it's too much without having wasted £££ or got an affiliated non-completion on your record might be a more achievable aim than doing a full season BE. Riding clubs often have friendly, low key dressage and SJ leagues which often are more suitable and affordable for people with other commitments than the same at affiliated venues - it's all about managing your expectations and priorities. That being said if you are serious about competing you might need to look at moving to a livery yard with facilities - you can do a lot towards fitness and schooling out hacking so it's not that per se but getting help with the horse and/or a sharer is a lot easier on livery than DIY IME...