OH is a HGV driver and qualified about 10 years ago.
He had to work agency for a while, as most companies wanted experience, but I suspect that companies will be less fussy due to the shortage of drivers. Even working agency, my OH has never been short of work and is constantly being offered permanent jobs.
Like you, we have a young family and OH has never had to work evenings or nights (unless he has wanted to). The type of jobs he has done, include home and office removals, the bins, food delivery, white goods delivery, scaffolding. All of these jobs have been done during the day. Drivers tend to start earlier, so 7am for example, to beat the rush hour, so I've done school drop off and OH has done the pick up.
The downsides - OH has to load and unload his truck, so physical strength is needed even with a pump truck.
The other downside is that he often gets sent down tight, narrow roads in unsuitable vehicles, so your DH will need to be confident in manoeuvring large vehicles. OH says sometimes, he has to reverse his truck back down those tight narrow roads because there isn't enough room to turn.
OH will say the biggest issue is traffic and he spends most of his time in traffic.
My OH is a natural driver (did about 5 lessons and passed his driving test first time). He has really good spacial awareness and can get himself out of the tightest of spaces. I, on the other hand constantly pranged my car when I got a bigger car because it took me ages to get used to a bigger size. I'd be a crap truck driver.