This video steps through WW2-era U.S. infantry tactical doctrine at the squad level, assembled from several contemporary manuals. In describing the final assault (@ ~22:10) on an enemy position (top centre) it outlines how smoke might be used to signal the fire shift by the base group (bottom right) from
for protection of the assault group (top left) as it closed. Implied but not explicitly stated is the expectation that the enemy would not properly recognize and respond to the fire shift for several seconds in the din of battle.
The video describes this technique, at the squad level, as one of several possible emergency measure rather than being preferred; but for a larger scale maneuver the obvious advantage of a smoke signal is better coordination over a wide area.
In his personal description of the island charge Maj. Richard (Dick) Winters notes:
Talbert, take 3rd Squad to the right. Peacock, take the left with 1st Squad, and I’ll take 2rd Squad right up the middle. Reis, I want your machine guns placed between the columns and I want good covering fire until we reach that roadway. Then, lift your fire and move up and join us. Fix bayonets and get in line as quickly as possible. Peacock, when everybody is in position, I’ll give you a hand signal and you drop a smoke grenade to signal our jump-off.
Good God! Right in front of me was a sentry on outpost, who still had his head down, ducking the covering fire from Lieutenant Reis.
This comparison of the televised scene to Winters' account differs slightly, mentioning:
On a smoke signal, the base of (machine gun) fire commenced, and the three columns started to move across the open field towards the Germans.
At this point, the Band of Brothers episode is deviating from the written version.
There was no delayed smoke signal causing the rest of the assault groups to wait.