Could lever action rifle have deserved to be used in military ?

At the time sure.
Faster than a bolt action.
Nowadays no.

Great for short engagements, where your 14 round magazine is enough.
When you the have to reload, popping a stripper clip into a bolt action rifle is simply faster than reloading the tube magazine cartridge by cartridge.

Lever action? No. Bolt action is more accurate.

Then there is the danger of using Spitzer ammunition in a tube mag.

NO. A repeating rifle, was not a problem with WW1. Nor was a .306 A .306 round, capable of severe damage. A .30 Cal round, was quite capable of killing. It’s the same reason a AK-47 is the world’s most destructive weapon system. 7.62×39 mm. A .30 caliber weapon, is indomitable. Especially at a high rate of fire.

The “bad guy” AR-15, fires smaller projectiles, 5.56 mm, or .223 caliber. Smaller projectiles, at a higher velocity, and a higher rate of fire.

There are vast differences between assault rifles.

Prone to jam & not good in a tight space which is why I prefer a pump action for family get togethers. XD

Yes. The lever action repeater is considered the first assault rifle.

The M-16 has several variants, one of which is a .30 caliber. A 5.56 round is OK for the jungle but not much good at penetrating cars, doors, walls, etc.

At the time the lever action rifle was developed, it was much better than the breech loading single shot rifles of it’s day. (Better to have one or two guys reloading rifles and the better shots doing the shooting.) Plus, the rounds for certain types of the repeater, could also be used in the Colt revolver, which meant a soldier could carry more rounds.
A bolt action may be more accurate, but in close quarters combat of, say, 50 yards, the lever action was accurate enough. And they were Winchesters, which is nothing to sneeze at.

Better than a Brown Bess

The M-16 was prone to jamming, in jungle, or desert. Men were found dead, with full clips. The piston driven design, could handle most environments. Another sad lesson learned, by war…

Lessons learned from the M-16 were used to create a better rifle, the M-16A1. It had a forward assist plunger to drive the bolt home if it jammed on cycling a new round into the chamber. Also the chamber was chrome plated that helped keep if from jamming. Also on the M-16A1, an improved flash suppressor and the 30 round magazine.
They have another variant now, the M-1616. Pull it out of the mud, shake it off, pump 100 rounds through it on full auto, and eject the bolt into your hand. It’s cold. It won’t burn you, which is one reason the hot bolt of the M-16 will cause a jam.