Why deploy any fighters at all if the Imperial Star Destroyers are bristling with laser cannons?
You can't hit a fighter with a capital ship turbolaser (which has a recharge rate of 1-2 seconds). Well, you can, but it's hard when the fighter isn't standing in place and there's tons of them. This is elaborated on greatly in X-Wing book series.
From C-canon, the Empire actually came up with the answer to this issue, in the form of Lancer class frigate, specifically to combat fighters.
Which brings up another question - wouldn't large capital ships be extremely vulnerable to swarms of fighters armed with torpedos?
They would. It was called by more brainy Imperial officers "Trench Run disease", named after Death Star I's trench run by Rogue squadron that destroyed it.
This would argue for fielding no capital ships (except as troop transports perhaps) and instead lots of small fighters equipped with hyperdrives & maybe some medium sized ships, eg the Millenium Falcon.
First of all, because the original fleet the Empire had was built before the effective space superiority fighter - coupled with superbly trained pilots like Rogue Squadron - were a known threat.
Second of all, because the Empire's fleet's main purpose wasn't to tangle with a fighter force - it was to subjugate systems and/or participate in large scale military conflicts (there are hints that Empire knew of Vong threat).
Once they did discover the weakness and the Rebellion rose in threat, the Empire did shift its tactics somewhat, to build up their own fighter force (Tie Advanced, Tie Interceptor, Tie Defender on the hardware side and 181st Imperial Fighter Wing and its likes on the wetware side) as well as anti-fighter ships (Lancer class frigates) and tactics designed to combine the benefits of capital ships (big firepower and carrying capacity that fighters lack) with supplemental parts of the fleet to defeat the fighter threat.
Out of universe, as you yourself noted, this was modeled on WWII combat, with capital ships and attack aircraft (both torpedo bombers and fighters to fight them off) replacing the era of battleships.