I don't have an attitude/intolerance condition, so I'll reply...
The paths in Petra are almost exclusively unpaved, though not too rough. There are hiking opportunities, but you can see some areas of Petra while staying on even ground. Getting close to most individual structures involves climbing stairs. Each of these obstacles is not impossible to conquer even if you cannot walk like a Tibetan mountain goat. But (and that is a big but) the archaeological site is huge. Most tourists (even lazy ones) end up walking several miles per day, often for several days in a row. So the challenge is the distance, not the terrain.
The carriages can only access a small part of the whole site (though it does include Petra's most spectacular parts).
If this is your only chance to visit Petra, use a carriage and enjoy the very limited view that you can get. If you have an opportunity to return in the future, use your own legs and see much more of Petra then.