I agree that he wouldn't use a loud voice if he's speaking to Edhellos directly. But if what he is doing is calling the attention of those around him to the fact that he's picked up Sauron's pet...he might have a reason to direct that comment to them, not her. (He probably wouldn't direct the comment to the
entire battlefield, of course.....)
So, hey, other balrog not-too-far-away, 'Look, it's Sauron's pet!' And then smash. What does he have to say to Edhellos herself?
If Angrod and Aegnor see him catch (and then kill) Edhellos, they will certainly try to confront him. But I don't know that they'd ever reach Gothmog themselves. Since we want
both of them to survive this encounter, they can't have much interaction with Gothmog, I wouldn't think. And Gothmog needs a reason to retreat unrelated to them (Fingolfin's arrival, as you said), so...what are we doing with that scene?
Corey Olsen strenuously objected to Sauron's involvement in this battle. He wants Gothmog to be the commander. He wants the troops (orcs and trolls) to be under Gothmog's command. He wants the strategy to be a simplistic hammer-blow, not a devious plan. (Two-different-waves not being all that devious, let's be honest.) He wants the failure of this battle to reflect poorly on Gothmog. He does *not* want Sauron to agree to serve under Gothmog, or be commanding Gothmog's forces (orcs). So...there's really not a reason to put Sauron on this particular battlefield.....and there's no opportunity to explain to the viewers what he's doing there if he does appear. (Sauron continuing his catch-and-release program independently of Gothmog's military campaign is fine.)
Rhiannon, of course, strenuously objected to Corey Olsen's objection

.
There is a reason for the balrog rule. Villains are less and less scary every time they appear on screen, particularly if there seem to be no serious consequences to the good guys for facing them. (Something that TV Tropes refers to as
'villain decay.') The balrog in
Lord of the Rings appears *once* - Durin's Bane can be very scary, and Tolkien builds him up well. While Tolkien occasionally has characters kill dragons without themselves dying, he
never portrays a balrog death as being non-fatal to the person who killed the balrog in his post-LotR writings. (Glorfindel and Gandalf got better....) Tolkien used the balrogs in great numbers in his early stories (a role he later filled with trolls; as the balrogs became fewer and fewer in number, leaving only 3-7 who ever lived in his latest writings on the subject. We certainly are going with the later version of very few terrifying fire demons, rather than a whole army of slightly-better-fighters-than-orcs.
So. Given that. We need them to be TERRIFYING FIRE DEMONS every time they appear in battle. They can't just be 'there.' They can't fight the good guys, kill a few extras, and then run off. They have to be devastating and make the audience truly concerned for the lives of the characters facing them. That concern will be diminished every time someone faces them and lives. That concern will not be significantly fed by the deaths of extras. Therefore...we've committed to killing off an actual character the audience has gotten to know and care about in the episodes prior to the battle whenever a balrog enters the fray.
Villain decay is not necessarily inevitable in a long-running TV series or series of films, but it's very likely, and even good writers struggle to avoid it. In
Buffy, the vampires are naturally difficult to kill...at first. They get a lot easier over time, so that while at first only a Slayer can handle them, eventually the slayer's completely human (well...) friends can handle it, too. Spike might be rather terrifying at first, but he eventually is mostly a joke (and later a semi-ally). How hard is it to kill a terminator in the first two terminator films? How about in later films/TV shows? Or an Agent in the original Matrix film, versus the 2 sequels? We are going to start killing balrogs eventually. Until then, we need them to come across as nigh invincible (though not actually bullet proof). And even when they die in the Fall of Gondolin...we still need them to be scary later. That's going to be tough to pull off convincingly. Let's not screw it up here.