Fast start can be excellent as you’re in the action and (presumably) getting an idea as to the story and style of game right off the bat and hopefully grabbing the player’s interest, but it can also be weird to have the player pick out actions if the game hasn’t yet asked if they’re X, Y or Z yet. It can also be a bit of a pacing issue if things go from “You stand over the smoking corpse of the Robo-Hellbeast, rubble scattered everywhere. You hear your mother calling your name, what was it again? Also, your brother and sister are going to call you ‘sibling’ for a couple more pages because that’s the soonest I worked in the gender choice.”
Drawback of a slow start is if you don’t do it right it can be hard for the player to get clear picture of the style of world and game they’re playing. Like, Choice of Dragon gives you the charging knight and ways to deal with it, and that gives a frame of reference for how other scenarios will be resolved, and the player can see from the options available what sort of different styles of play/stat builds might look like (I’m a fire mage vs. I burn them to ashes with a gout of flame from my fingertips). On the other hand if you provide too much world-building information it can come across as info-dumpy and the player might not be able to discern what’s relevant to their play-through and whats background fluff to flesh out the world.
Personal preference I would lean towards a fast start, but in the end it doesn’t matter to me which one is used so much as whether it’s done well. Choice of Heroes certainly has a fast start, but it doesn’t end up being relevant to the rest of the game at all other than to sort of broadly establish the technological level of the world, which was done better by the mechips anyway (And to be clear, I overall liked CoH despite its problems)
edit: Wow, too many words. TL:DR; there’s pros and cons to both, I like fast-paced