The lever action is uniquely "American" it was designed here and flourished here, although it has made an impact overseas (the Russians were massacred at the Battle of Plevna because the Turks had Winchester 1866s in addition to their single shot rifles, so the sudden appearance of a wall of guys with 14-shooters vs single shot Russian rifles= death for lots of Russians).
They are designed to be light and handy. They are. Shooting on the move or from horseback (I've heard, haven't tried it on horse), and making quick snapshots, are easy with a levergun and sometimes more easy with one than a bolt gun.
Their design is well balanced and good to carry all day in the hand without being encumbered.
Although a good bolt action man can equal it, without using a semiauto, many people are faster with a levergun than a bolt. Your hand can't slip off the bolt handle as the lever has your hand "in" it.
Older levergun designs need flat-front bullets from the tube magazine, but the Savage 99 (1899!), Winchester 1895 and the newer Browning BLR all had/have box magazines to fit modern rifle rounds and don't give up the levergun's other attributes. Teddy Roosevelt loved his '95 in .405 Winchester as he had a solid hitting hard use rifle that was fast and accurate.