I have no reason to believe that lever guns are inherently more or less reliable whether the cartridges are rimmed or rimless. Older designs like all pre1895 Winchesters, and the Marlins you reference, were all originally designed for rimmed cartridges because that's what existed back then. The rimless cartridge didn't appear until Mauser invented it in 1888, and even then it took a few years for it catch on.
The rimless cartridge made the use of a box magazine possible, because you didn't have to worry about the rims hanging up on each other. That in turn meant that you could use sharp-pointed bullets like spitzers, which you can't use in tubular magazines. As a result, lever guns that are designed for more modern, rimless cartridges generally have box or rotary type magazines (examples being the BLR, Savage 99, Winchester 1895, etc.).
As far as I know, you could easily use rimless cartridges with a tubular magazine, lever action rifle. You just have to use flat pointed bullets, and most modern rimless rifle cartridges use pointed bullets.
