Where is Aeglos?

[Gordis]

Actually, the only Elven horse we see up close - Asfaloth - did have a saddle and stirrups. Thus Glorfindel could use a lance in a normal way. Why not Gil-Galad?

But there is another thing I remember: each of the peoples of the Eldar were associated with specific weapons:

Morgoth's Ring; The later Quenta Silmarillion:
Other names in song and tale are given to these peoples. The Vanyar are the Blessed Elves, and the Spear Elves, The Elves of Air, the friends of the Gods, the Holy Elves and the Immortal, and the Children of Ingwe; they are the Fair Folk and the White
The Noldor are the Wise, and the Golden, the Valiant, the Sword-elves, the Elves of Earth, the Foes of Melkor, the Skilled of Hand, the Jewel-wrights, the Companions of Men, the Followers of Finwe
The Teleri are the Foam-riders, the Singers of the Shore, the Free, and the Swift, and the Arrow-Elves; they are the Elves of the Sea, the Ship-wrights, the Swanherders, the Gatherers of Pearl, the Blue Elves, the people of Olwe.

So, a spear is a weapon assciated with the Vanyar. Gil-Galad was at least partly Vanya (through Indis), maybe he had more Vanyar blood: because we don't know, I think (?), of what people were Angrod's and Orodreth's wives.
Anyway, it may well be that a Vanyarin lance had been brought to ME by Fingolfin and Finarfin's people and later passed as heirloom to Gil-Galad?

I agree with Mithadan: what remained of Aiglos would likely be sent to Valinor to await G-G's reincarnation.