If you're going non-fantasy (in which case you can put in whatever), I think that one factor is also that in, say, the Napoleonic era, using soldiers in formation in warfare was an important multiplier, and that's not super-friendly to FPSes. I mean, a lot of the game would be following orders to move into a formation or move in formation.

As for weapons, you could do archery, I suppose. There have been a number of games (Thief, Skyrim, etc), that have an archer running around on their lonesome, though that probably wasn't historically all that accurate. Well, not that having a solo character going Rambo on a World War II-and-post battlefield was necessarily all that common. If it did, it was pretty unusual:

https://en.wikipedia.org/wiki/Joe_Hooper_(Medal_of_Honor)

For conspicuous gallantry and intrepidity in action at the risk of his life above and beyond the call of duty. Staff Sergeant (then Sgt.) Hooper, U.S. Army, distinguished himself while serving as squad leader with Company D. Company D was assaulting a heavily defended enemy position along a river bank when it encountered a withering hail of fire from rockets, machine guns and automatic weapons. S/Sgt. Hooper rallied several men and stormed across the river, overrunning several bunkers on the opposite shore. Thus inspired, the rest of the company moved to the attack. With utter disregard for his own safety, he moved out under the intense fire again and pulled back the wounded, moving them to safety. During this act S/Sgt. Hooper was seriously wounded, but he refused medical aid and returned to his men. With the relentless enemy fire disrupting the attack, he single-handedly stormed 3 enemy bunkers, destroying them with hand grenade and rifle fire, and shot 2 enemy soldiers who had attacked and wounded the Chaplain. Leading his men forward in a sweep of the area, S/Sgt. Hooper destroyed 3 buildings housing enemy riflemen. At this point he was attacked by a North Vietnamese officer whom he fatally wounded with his bayonet. Finding his men under heavy fire from a house to the front, he proceeded alone to the building, killing its occupants with rifle fire and grenades. By now his initial body wound had been compounded by grenade fragments, yet despite the multiple wounds and loss of blood, he continued to lead his men against the intense enemy fire. As his squad reached the final line of enemy resistance, it received devastating fire from 4 bunkers in line on its left flank. S/Sgt. Hooper gathered several hand grenades and raced down a small trench which ran the length of the bunker line, tossing grenades into each bunker as he passed by, killing all but 2 of the occupants. With these positions destroyed, he concentrated on the last bunkers facing his men, destroying the first with an incendiary grenade and neutralizing 2 more by rifle fire. He then raced across an open field, still under enemy fire, to rescue a wounded man who was trapped in a trench. Upon reaching the man, he was faced by an armed enemy soldier whom he killed with a pistol. Moving his comrade to safety and returning to his men, he neutralized the final pocket of enemy resistance by fatally wounding 3 North Vietnamese officers with rifle fire. S/Sgt. Hooper then established a final line and reorganized his men, not accepting treatment until this was accomplished and not consenting to evacuation until the following morning. His supreme valor, inspiring leadership and heroic self-sacrifice were directly responsible for the company's success and provided a lasting example in personal courage for every man on the field. S/Sgt. Hooper's actions were in keeping with the highest traditions of the military service and reflect great credit upon himself and the U.S. Army.[4]

That's a pretty unusual MoH citation out of Vietnam, and that'd probably be about par for the course for a single -- maybe part of -- a WW2 FPS level. I mean, if you want realistic World Wars fighting, the largest chunk of characters would probably just be killed by random artillery fire, not pulling off 100:1+ kill ratios in infantry combat, which...isn't all that much fun as a first-person game.

But, as to archery:

https://www.tastesofhistory.co.uk/post/dispelling-some-myths-archers-shooting-twelve-arrows-a-minute

A skilled longbowman could shoot about 12 shots per minute. This rate of fire was far superior to competing weapons like the crossbow or early gunpowder weapons...

So, as to the hail of arrows, archers shooting heavy warbows confirm that releasing twelve arrows in one minute is possible, but that such a rate cannot be maintained subsequently. Practical experience argues for a shooting rate of about 5 to 6 arrows per minute being feasible over a period up to 10 minutes.

That's definitely a lot slower-paced than a modern FPS, but it's still a lot faster than nearly all 18th century firearms.

Skyrim kind of ignored fatigue and let you lug around a huge store of arrows and blast them without regard for your arms getting tired, so it's not hard realism, but I think that people enjoyed the archery aspect.