My 13YO (at the time) son and I rode to Paris, gave the bikes away (as part of the plan) and flew home.
I rode a ten speed Raleigh Pioneer, that I found dumped in a hedge in pieces. I reassembled it, regreased the hubs and spent £2.50 on a tyre. My lad's bike was £11 on
ebay (Saracen MTB with seized suspension forks). That needed a rear tyre changing which cost me £3.60 and a rack which cost £2 from the dump.
Mechanical problems on the way consisted of one puncture on my bike and, err, that's all. We had a brilliant time.
So, in my experience, it's perfectly possible to tour on crap old bikes. It's only a couple of hundred miles in France, not a trip across the Sahara. If everything collapses in a heap, just get on a bus/train/plane and come back home.