Trying to apply one challenge concept across different games, particularly in different genres, just doesn't work that well. For example, initial equipment works fine for an RPG, but go try playing a Grand Theft Auto game with the initial equipment rule... =P
For Lemmings, we already have a pretty damn good selection of challenges to pick from, and plenty of levels to do it on - a lot of the challenges don't cover even the Xmas/Holiday lemmings games, let alone other games that are mostly based on the original Lemmings formula (other-platform versions of Lemmings with unique levels (PS3 comes to mind), Lemmings Revolution, you could possibly argue L3D somewhat fits the definition (although even in the lower difficulties, L3D levels are often built to a standard of "give the player exactly what they need" and, aside from a few levels - including the final one! - having horrific backroutes, it's generally one solution per level)), and even when those run out we have plenty of fan levels to turn to - there's a decent number of packs comparable in size to the original games (the Lemmings Plus packs, PimoLems, Revenge of the Lemmings) and then all the small packs or individually-released levels... we pretty much have plenty of ways to challenge ourselves as it is.
Though the idea of going through the original games losing as few lemmings as possible is an interesting one...