Yes, for one on one.
Spear alone dominated sword alone.
Spear alone narrowly beat sword+shield.
Spear+shield tended to lose to sword. The spears lost too much reach, speed, control, and power when used one handed, at least by inexperienced users.
However, they did try a formation of spear+shield, a rough shield wall, and in formation it appears the shield and spear combination is the way to go.
On the skill vs. courage for swords, the Roman fascination with martial virtue and ideals may play into that, and both explain their heavy use of swords and success with them. They seemed to try to be trained and disciplined soldiers and mad berserk warriors all at once in a way. Psychology of that must have been interesting... with rather a lot of inner tension that they seem to have then externalized in various violent ways...
My memory on this is vague, but I had in mind the attempts at lance use by the USA and CSA was distinctly unprofessional by European standards. Union cavalry once it finally got sorted tended more towards a mounted infantry or dragoon model, while the CSA did some of the same plus tended toward something more like light cavalry with particular skill in scouting, raiding, and ranging.
US cavalry spent a lot of thought on fighting against lance armed cavalry (Indians, Mexicans, Europeans) but seems never to have made much effort on sorting how best to fight as lance armed cavalry itself. Poor showing with lance is thus rather expected I should think.
The biggest drawback of the lance is that it is long, bulky, heavy, awkward, etc. It is a pain to carry and it has to be discarded to use other weapons. Saber and pistol were more versatile side arms, and convenient to carry. Even a carbine could be holstered or hung from a saddle easily compared to a lance.