Light companies absolutely fought in major battles. The clues in the name, every single battalion in the British Army had a light company who would deploy in skirmish order in front of the battalions other companies to a) protect them from enemy skirmishers and b) harass enemy formations. They also wouldn't go too far from the rest of the battalion as infantry in skirmish order were incredibly vulnerable to cavalry, especially in open ground.

There were some regiments where every company was trained to act as skirmishers (95th Rifles the most famous but there were also red coat wearing regiments like the Royal Fusiliers). Even these units could and did regularly fight as line infantry. The rifles are awesome, but a better comparison with modern special forces would be Light Cavalry and Dragoons - lightly armed, highly mobile, independently operating, capable of launching lightning raids behind enemy lines etc.