Los Navas de Tolosa had a serious French contingent, from what I know, as did many other battles in Iberia. I am sadly no expert, but I found the book: El Cid and the Reconquista 1050-1492 By David Nicolle. It says outright there was strong French influence on the reconquista.
David Nicolle, or René de Beaumond (Les Croisades Franques d'Espagne) indeed make a point. But, the french influence is not as you describe, made by
adventurers, nobles on the loose after they didn't warred in their own land.
Using the exemple of Las Navas de Tolosa, the "french" participation was real, but limited. Arnaud Amalric, bishop of Narbonne, was present with many troops as
vassals of Pere II, not as adventurer.
Admittedly, you had as well "crusaders" from Europe (as the pope made campaigning against Almohads beneficing from the same features than a crusade), but that was quite exceptionnal and obviously didn't played before the XIIIth century.
It is hard to estimate how important was the part of the army that was raised in France, critically when they didn't all participated to the battle (both by lack of logistic, and being a nuisance in Spain)
The sources exaggeration don't help at all (mentioning 100 000 warriors, something really not possible). Considering that southern France was currently in full war (meaning more important vassals couldn't join), I would say 3 000 to 5 000 (including ones not joining the battle) would be an appropriate guesstimate.
On an army counting 12 000 to 15 000, it's fair but not that decisive.
Al-Andalus was certainly rich, so perhaps France alone wouldn't have been enough; I can't find very accurate numbers on numbers. Although I've always heard Aquitaine/Occitaine were very rich too until the Cathars got wiped out...
That's not really comparable.
First, Al-Andalus height was passed at 1209, with the taifa crisis and the Berber dynasties decline.
Then, Aquitaine/Occitania even at his political height didn't had nearly the economical influence of Al-Andalus. One exemple among many others : "imposing" a monetary standard (Carolingian silver coins being quite similar to Al-Andalusia's).
Summarizing it, southern France was wealthy. Al-Andalus was
rich : cereals, sugar, wine, silver, oil, tissues, fruits, gold, wool etc. (and I'm not even talking about trade reaching both Germany and Persia). The only things they lacked were iron, and less importantly wood.
Still, the story from Tours until 1212 or so seems to be one of western christianity ascendant
It's mainly an historiographical conception : while Arab conquests were the fact of a relativly unified command; western Christian advances were mostly local from Xth to the XIIth centuries, and the fact of separated polities.
While it show an efficient politico-military system, it wasn't unbattle or prevented counter-attacks. For instance, you had ongoing raids in Gaul up to the XIth century, mainly trough Provence up to effective control of
provencal coast (and with effective control of Alpine passes as well)
Rise of western Europe isn't exactly the tale of an unrivaled growth.