First of all, its not an either or situation.
Now, if you are asking what does a lever gun bring to today's market that other guns don't? Well, lets take a look....
The classic and most common centerfire lever guns are the Winchester, Marlin and Savage 99. The Browning BLR is a realtive newcomer, and no where near as common as the others with their over a century of production of different models.
All these rifles have some things in common, besides the lever. They are flat sided at the balance point (the action) and fit well in a single hand carry. Some bolt guns fit well, as do some pumps. Very few semiautos do.
Well, an AR15 chambered for 6.8SPC with an Aimpoint is better, but it also costs more.
One thing you'll have a tough time doing is carry that AR comfortably balanced in one hand...especially if its a flattop model.
Another thing lever guns bring to the market is the range of cartridges available. The BLR will handle modern high intensity rounds, matching most of the bolt gun's ablilties there. SO, if you are sold on a lever, you need not be stuck with a turn of the 20th century round, (which still does the job quite well enough) if you don't want to be.
There are models of lever gun made to shoot pistol rounds. Not the 9mm everyone loves today, but .357 Magnum and .44 Magnum, as well as classic old time rounds like 32-20, 38-40, 44-40, etc.
There are some semi autos that handle 9mm and .45ACP, but many factors work against those rounds as best for small game, and out of carbines they are not the best available choices for deer even when only compared to other "pistol" cartridges.
There are no bolt guns, or pumps in common use that have actions scaled to the pistol cartridges. They are all built to take the rifle (2" or more) class cases.
Another thing a lever takes advantage of is that in suitable calibers, they are legal everywhere. There are some places where semi-autos are not legal for hunting. No advantage over the bolt or pump there, but if you live where semi's are not allowed that takes them completely out of the running.
Another point for the lever gun, is magazine capacity. Matches the bolt & pump, at worst, and in some calibers exceeds them. There are also lever guns with detatchable magazines, so if you wanted to, you could match the capacity of the common semis, game laws permitting.
Rifle weight can vary widely, some lever guns are quite heavy (heft an 86 Winchester) but the common Win 94 is pretty light, and the Marlin 336 only slightly heavier. The majority of bolt guns tend to run a pound or so heavier.
One thing to remember is that bolt guns, and the majority of semis are adaptations of military arms. Some of them are quite drastically evolved from those origins, but those are the origins. Rugged, and generally heavier constructed than absolutely necessary for sport hunting. Generations of hunters took off wood (and often metal) from military bolt guns to make them better adapted to sporting use. Semi's (until the most recent generation of designs) were even less adaptable, partiucularly in terms of removing excess weight.
In today's market? Lever's don't offer a lot that other actions can't do, but there are a few things that they still shine at, just as well as they always did. No rifle design is the best at all things, and what's best for some things is far from best for others. As a class, there is a wide range of capabilities available from lever guns, for for millions of hunters over the years, plenty to bring home the venison.
Pick any reasonable (North American) hunting situation, and you can find some lever gun that will perform more than adequately well (if the shooter themself is up to the task). You can also find a lever gun that is not the best choice for a given situation. Pretty much the same as other action types in that regard. Remember what they were made for, small and big game hunting. Not combat, not benchrest, or precision varminting. A good enough, if not the best possible for most things, and damn good at a few things.
You can think of lever gun guys as John Wayne wanna be's if you want. Doesn't matter to me. For a lot of us, old AND young, there is more to life than alloy and plastic. There is steel and walnut!