Since we are freely speculating, let me add some of my own
Sending Glorfindel along (though it might have been considered by JRRT as the suggestion is hinted at during the council) would tie the story to much with the Legendarium - and yet Tolkien was writing a
new story - descended/coming/arising/ from tied to/hanging on the past, but not being the past in itself, if you follow my meaning) would have been a breach of certain general canvas:
Mark like all great characters from the past are passive - Cirdan, Elrond, Galadriel (the latter, admittedly, 'striving with the Enemy in her mind' (excuse lopsided citing, I'm on the go, citing from memory) - but they don't move about much - their purpose is just that - to tie the current events with those of first and second ages
They are all tied to certain places too (resembling Tom Bombadil in that respect) - not going far from their own boundaries that they themselves have set (well, Aragorn-Arwen and Elrond at the wedding is another story). When Glorfindel roams whereabouts of Rivendell in search of Ring-Bearer and the fellowship, he's actually doing as much as all them 'old chars' do - 'beating the bounds', or going just little beyond the boundaries of their own realm
Whilst sending Glorfindel along would turn the story into something resembling the first age (and also diminish respective roles of all others. + I don't thikn Gimli-Glorfindel friendship line would work as good as Legolas-Gimli line did. For dwarf-elf controversy, goes back to first age too, with Thingol's murder and silmarils, but immediate connotation here is restoration of friendship between Erebor and the Forest because of what Thranduil did to Thorin and Co (and of course Gloin Gimli's father)
One might argue here that Gandalf, being Olorin, is first age person too (Saruman, Radagast and Sauron too, for that matter)
But Sauron acts just like the other 'oldhands' - sitting in his realm, just being the baddie and the bully he's sending his chaps along to conquer (but himself, he's tied there too - striving with Galadriel in mind (the other side of the same coin) and roaming the world with his Eye)
Whilst Istari are cardinally new thing - old spirits though they may be, they are incarnate
I hope I'm coherent enough, and did not tire you with this stream of consciousness - as I'm almost thinking aloud, writing it down as it comes to my mind
Cheers