Yes, the H-harnesses are designed for lap-belts. If there is a correctly fitting lap-sash belt then use that instead.

I think the point of the booster is two-fold - 1) to put the kid's bum in the optimal spot for correct fastening of the seatbelt (ie right back against the seatback) and 2) to position the seat belt best for a child (eg some of the boosters have slots for the seat-belt to run through). I read the guidelines the other day and the message I took away from it is to look at your child's body, especially the length of their legs, if they can sit back against the seat and the edge of the seat comes to behind their knees, then they're big enough to forgo the booster (in that they are now big enough for the seltbelt to work correctly). Of course if you've got a tall one with flaps around the head then that has a protective role (side impact crashes) but that doesn't apply to the cushion-style boosters.

IMO those cushion-boosters aren't very useful - my eldest DD went into one of those when she got too tall for the other booster seat, but then it was less than a year until her eyes were higher than the seatback (ie too tall), so then it was no booster. From memory that was around 5 or 6, but that was back in the bad old days when there weren't any other bigger options on the market. We've now got on extendable one with the side-of-head flaps and I intend to keep DD2 in it as long as possible, based on her physical size. I suspect 10 years old is the absolute utter limit I could stretch it out to, but in reality, I have tall children so it will be before then, based purely on size.