That works if the role of the precursor beam is just to guide. If it is supposed to create a conduit, akin to the ionization path along which a lightning bolt travels, then moving the precursor "tag" means having to create an all-new path.
Klingons love to walk their pulses to a target, not wasting time and effort with making the first shot hit. The Defiant
does that a lot, too. For some reason, Starfleet beams
never do this, though: if they make contact, they stay in contact, but if they miss initially, they won't acquire contact later on, either.
(We see an extremely rare example of beam walking in "Jem'Hadar", as the runabout tries to hit the Jem'Hadar ship on a suicide run - and fails to connect the beam, despite swerving it rather desperately. Practically all other beams in Star Trek hit their target and stay with it.)
Are there other weapons we did not get to see as the action was so heavy and fast?
This is a good question, as dialogue doesn't necessarily specify which weapons are in use. We're still debating exactly what the Kelvin
was firing in the previous movie, as the dialogue only associated her red beams with phasers but never mentioned the white pulses...
We know Marcus wanted "torpedoes" targeted on the Enterprise
and got his giant gun barrels pointed at the starship as the result. We don't know the names of the other weapons seen. But we never really saw any other weapons besides these "curveball" red things, and there was loose talk about "phasers" there, so those two probably have to be paired in final analysis.