They all work fine and have their proponents.
Free spools typically have more capacity and are lighter then egg beaters.
Egg beaters/fixed spools typically have stronger drags, easier to cast and more forgiven when used on poorly designed rods.
Side caster/alvey have massive capacity, are quite inexpensive, are very basic in their design so easy to maintain. They typically have a 1:1 gear ratio. Never seen a Rod designed for casting with a alvey reel so modifying a existing one would probably be required.
For the type of fishing that is done on our east coast beaches it mainly personal preference.