[Personally I think the battalion would probably fight just as well whether their Brigade Commander was 10m away or 500m away.]
I agree. And for the most part, they will. But what they *won't* do is:
1. Charge the enemy on their own initiative without having been ordered to.
2. Change out of the formation he put them in, unless they're faced with an existential crisis.
And (this is admittedly an assumption that might not hold up in all cases, but probably often) they're probably out on their own somewhere, like the "Hold this village!" situation you described, without support to the flanks or rear, meaning they're arguably a bit shaky.
A few years ago when I was researching my Schill book, I walked the battlefield of Dodendorf with the local town historian. I wanted to see where those two Westphalian battalions had been when Schill charged and broke them. The position was... okay. They had one flank protected by a wooded stream, but the other flank was hanging in the air. Most importantly, though, they were on their own. The brigadier (von Uslar) had put them in a line covering an open field, even though he knew that they were likely about to face enemy cavalry led by a famously charismatic and aggressive commander (Schill). Then he left them there and went back to the town to talk to his French colleague (who had infantry in the village, and a half-battery of artillery.)
Guess what: Schill chooses that moment to charge. The Westphalians just stand there in Line, pop off a few shots at the oncoming Hussars, and then run like Hell. Schill broke both battalions - a force that outnumbered him 2-1 - in a couple of minutes. The Westphalians never tried to form square, never tried to fall back toward the town, and certainly never attacked.
In "Lasalle Terms" that combat would have looked like this:
1 Large Prussian Hussar unit (Valiant, Experienced), led by a brigadier with a * Bonus
charges simultaneously into:
2 Westphalian Infantry BNs (Shaky Amateur), presently out of command
The Westphalian player would try to React into Squares, but he'd need a 5+ in each case: his normal 4+, but he's out of command. He fails both rolls.
Schill picks the first Westphalian defender: The Westphalians have 8 dice, -2 for Out of Command, is 6, halved for not being in Square, equals 3. Schill's Hussars have 12, plus 4 for two levels of higher Esprit, equals 16, halved to 8 for an attacker versus two defenders. With his * Bonus, Schill's boys hit on 4+.
So the Westphalians have 3 dice, hitting on 5+. Schill has 8 dice, hitting on 4+ . Splat! As the Westphalians run away, Schill's Hussars wrap around the remaining battalion and do an even worse number on them.
Sorry to bore you all with this little story/example, but I think it tries to capture what I mean when I say a unit is "Out of Command."