I actually ended up taking three trips in the lift.
I climbed the stairs and was absolutely exhausted when I got to the display level. I saw a lift and asked if it was to the roof. Got a few positive responses. The lift duly arrived, we all piled in and it went down! I'd apparently joined a small guided tour that had lift access.
I asked the guide if I could go back to the top because I'd made a mistake thinking it was going up. I was quiet, polite and must've looked very sad and exhausted. She told me to wait at the entrance to the lift while she got her next group. A couple of minutes later I was heading back to the top and then found the stairs to the top where I enjoyed the view.
I then took the lift down at the end of my visit. It was a great view from the top of the Arc de Triomphe and I would recommend it for a great view of Paris and the massive round about that circles the Arc.
I did see a couple of stupid people run across the round about. I used the tunnel underneath to get there safely.