While I agree, under some circumstances, with your second clause, I couldn't disagree more with the first.
Second place to fun? Flavor is the fun. Mechanical goodies are nifty, but without a cool character, a cool setting, cool description, it's all just math. (Obviously I'm speaking for myself and the people I game with, not for every gamer ever. But I do feel strongly about it.)
I've heard people say this sort of thing a lot, and I think it's a real half-truth.
My
personal experience is that, in-game, if you have a character who is, thematically/flavour-wise "cool", as you put it, but who is in actual practice, ineffective/mediocre/bad-at-what-he-is-supposedly-good-at, then that actually curdles the "flavour"-fun. Nothing like having a PC who is supposedly great at unarmed combat (fr'ex), but who proves, due to the rules, to be a pretty bad combatant (see this in a number of systems). It's disappointment in it's purest form.
Similarly with "cool description" on powers and the like - if the flavour for a power is great, but in practice it's pretty worthless or doesn't really do what it says, then that, in my experience, leads directly to disappointment and disillusionment. Further, players tend very much to blame the game/system. They're not stupid. They know that it's not them, or the DM, or the adventure that made this thing suck, they know it's the rules attached to it. I've heard players say that they do not want to play specific systems for reasons that are essentially flavour/mechanics mismatches plenty of times (it's probably one of the most common "Ugh let's not play that!" reasons, after "The rules are too complicated!" - which is admittedly the most common reason by far - in my experience).
I've personally experienced it a number of times, especially when I was younger, where I've had an awesome PC concept,
seemingly supported by class or setup in the game, only to find, actually, that just doesn't work very well, or worse, that class literally can't do the thing it's described as doing*. This is a problem even if the class is actually very good at something else, because it's not what you signed up for, it's not the fantasy you wanted to live out. Indeed this is one of the big reasons I got interested in actual mechanics and how they function in practice.
So flavour absolutely is a significant part of fun - but when flavour and mechanics mis-match, that can actively harm fun more than if there was no or weak flavour (if they align perfectly, that's awesome, of course - some of 5E's classes/subclasses certainly do - but only some - I'd actually say the October playtest
generally had better flavour/mechanics alignment).
* = Bard in 5E is an example of this - they can't support the party in the way the cool flavour-text describes - simply don't have the abilities for it (they easily
could have, with a couple of songs-as-spells, but WotC didn't bother). Rogues were often victims of this in pre-4E editions, where they're often written up as fearsome killers, but actually, even when the stars align, they're only keeping pace with people making far less effort (4E fixed this, 5E looks to be close to 4E rather than previous editions on it). 4E certainly had plenty of problems in this department, though, especially in the awkward PHB1-2 period.