Fuel – Latin or Teutonish?

Today’s word of murky upspring is fuel.

The Oxford Dictionary Online (ODO) has:

fuel, from Old French fouaille, based on Latin focus ‘hearth’ (in late Latin‘fire’)

Online Etymology has:

fuel, early 14c., from Old French foaile "bundle of firewood", from Vulgar Latin legal term *focalia "right to demand material for making fire", neuter plural of Latin focalis "pertaining to a hearth", from focus "hearth" 

Wiktionary gets a little nearer with:

fuel, Old French fouaille, from feu (“fire”) … but then goes down the same path with feu from Latin focus (“hearth”), replacing Latin ignis (“fire”)

Rooted on focus? I think not. On the Teutonish word for fire? Much more likely! 

Let’s see what Kluge has to say about the root of fire: 

MidHG. viur, OHG. and OLG. fiur, older fuir, n.; comp. Du. vuur, AS. fyr (from *fuir), n., E. fire; a word common to West Teut. for ‘fire’: … comp. OIc. … furr, m., and fyre, n., ‘fire’. The r in all the words is a suffix, and fu … the root … p86, Feuer

We know that the French -aille is in English -al or -el. We also know that the French ou is a ū sound as in vous. The Teut. root is fu. Thus the French spelling of a Frankish/Teutonish root for fire would be *fou or *feu (See German Feuer). Now add the ending -aille and voilà: fouaille.

There was no need to twist and bend Latin focus into feu when there was already a Frankish/Teutionish root fu.