I've decided to post my experience in case it's helpful to someone down the road. The bolts did, in fact, run through the floor. They were lag bolts set in floor board, then covered by a double layer of floor board with the vinyl stair material glued on top. First, I had to peel back the vinyl covering. It took some muscle but the adhesive was still tacky. I also had to remove the backing from the next stair up and pull out the carpeting on the left side to make room for the floor board to come up.
The floor board was glued and secured with many, many staples. I appreciate Newmar's build quality, but I'm not sure where they thought this board was going to go. All those staples posed a real challenge because they were impossible to dig out individually and there was nowhere around the edge of the board to insert a pry bar. I ended up putting two screws in the top of the board, attaching a wire to the screws and using a long piece of metal as a lever to pry the board straight up.
Here's a picture showing all the staples. There are at least 20 of them and they're much longer in person.
When I put it back down, I'll secure it with screws in case (God forbid) I ever have to do this again.
This exposed the underflooring with the lag bolts set into the wood.
There was no way for me to get a hold of the tops of the lag bolts so, once I saw what I was up against, I used a grinder and a dremel to cut the four nuts off from the bottom.
The wood is in decent shape but isn't strong enough anymore for new lag bolts to gain a purchase so I'm going to replace them with washers and hex head bolts.
This is proof that the stairs really do come off!
Hopefully the reinstall goes easier. I'll post here if I run into any unusual challenges.