If I was going to point out the deficiencies of 4e for narrativist play, I wouldn't look at the mechanics of the PC build rules at all. The two main things I would point to are:
(1) That it is possible, within the rules, to build a PC who has no significant thematic hooks. A halfling archer-ranger of Avandra or Melora is in danger of being such a character. For the designers, I think the fact that some options are thematically more vanilla than others (eg dragonborn, tiefling, dwarves, warlocks at the strong theme end; rangers, halflings, wizards towards the other end) was probably a feature. But for narrativist play it is a bug.
(2) That it is possible, within the rules, for the GM to frame encounters that have no significant thematic bite. Kruthiks and ankhegs are in danger of being such encounters. The rules have good advice on ensuring tactically engaging encounters, but are pretty weak on the thematic side of things.
Happily, both these issues can be avoided by making judicious choices from the list of options open to players and GMs.
A third criticism that might be run is that the actual thematic span of the game is not very wide, but then the same could be said of MHRP. This doesn't make it poor for narrativist play, it just means that you don't pick it up for that purpose unless the thematic stuff you want to engage with is stereotypical heroic fantasy.