Yes, they will die quicker from crossbows, however right now tartar steppe lancers aren’t even doing terrible against chocked up cross bows when compared to cavaliers. They were quite comparable actually, though my tests were not perfect and I might be wrong about that.
So what you want to do is to give them even more pierce armor, reducing the upgrade cost, give it bonus damage against villagers AND decrease the overall cost? I think with those stats it would become way too strong again. Maybe it will need more than +1 pierce armor, maybe 2 base, 3 elite is necessary. With 3 pierce armor as elite(or tatar lancers with 2 pierce armor as elite) they would soak up 0.3 arrows per ressource. Cavaliers take 35 shots which is 0.26 arrows per ressource (15% more cost effective in tanking). With elite 3 pierce armor the tartar one would then take 50 shots, so 0.44 shots per ressource, that would be the same as the Paladin. So yes, compared with the additional range I think they would find their niche as anti archer cavalry.
So I think either: +1 pierce armor through elite upgrade and reduce cost of elite upgrade, or +1 baseline +1 elite upgrade and keep the cost of the elite upgrade.