Yup. I've seen guys take on side jobs like you're describing and cut lines, pipes, one guy even dug up a grave that was an old cemetery. I had one customer talking about how he was bush hogging for a guy and hit a piece of an old tire, and sent that piece of tire through another guy's house. Insurance would've covered it but he didn't have any, so the homeowner sued him and easily won. IIRC it was about $4000. Guy who was doing the side work admitted that he should've had insurance.
Thing is, if you're insured, you've got to charge a little more to offset that cost of the insurance. BUT...if you can advertise that you're insured, it may draw in more work. Friend of mine bought a brand new L35 in 1995 and went into grave digging as a main job and then side jobs doing plumbing lines, etc. No insurance. I mentioned it to him and he agreed that it'd be a good idea and went ahead & bought a policy. Advertised it too. He was as busy as he could stand to be from then on..and I mean REALLY busy. Paid the tractor off in 13 months.
I even know of a dealer who loaned a dealer-owned "loaner" lawn mower to a customer...as I recall it was a conventional riding mower. The mower had the seat switch bypassed, was that way when the borrower picked it up. Anyway, the borrower cut the yard with it and then got 3 of his buddies rounded up. Mower was started and blades engaged, then the 4 guys picked the mower up by the deck in an attempt to trim the bushes. Yep...lots of cut fingers. Now this was a weird deal for sure, but the 4 guys SUED the dealer for about a hundred grand....and WON; on a technicality (seat switch was bypassed at time of pickup and nobody bothered to tell him). Point being, you never know what you're up against...but if you try to protect yourself as best you can generally speaking you'll be ok.