Fireteams are nothing to do with Battle Drill. Battle drill was started in 1918 using entire platoons, not sections. If you can find a copy of Maxse's and Liddell-Harts 1919 manual, (SS-148??) it shows this in detail.

What the US adopted in the 1950s is nothing like the UK iteration of Battle Drill.

Battle Drill always talks about attacking from the rear or flank, or even by-passing. The 1942 Instructors hand book is explicit on this. Frontal attacks were used in training because of live ammunition range safety templates. How you train using live fire is still a major consideration on this matter.