The fun way to do this, as well as the obvious interpretation of the rules, would be to have them imitate phrases perfectly, sounding like the original speaker, but using them in approximately the sense they originally heard them. So their voices can change completely from sentence to sentence - one being a dwarf they heard speaking yesterday, the next sentence being a small elf child they heard ten years ago, and the third being something you said to them three months ago.

They understand what phrases and sentences mean, but if they can't break down phrases into words and use the words individually, they're still fairly limited by their curse. They would sound a bit like Dilbert's pointy-haired boss, who tends to come out with phrases and clichés that belong in a subtly different conversation.

This would be an interesting challenge for a player who was usually verbally adroit. Precedents for even more extreme versions exist in the Librarian from Terry Pratchett's Discworld series who manages to get by with just one word, "Oook", and Mr Punch in the classic Punch & Judy show, most of whose lines are "That's the way to do it!"

The main problem with playing a Kenku this way would be avoiding humour in serious moments of the game.

No

Player's Handbook page 7:

If a specific rule contradicts a general rule, the specific rule wins.

... For example, many adventurers don't have proficiency in longbows, but every wood elf does because of its racial trait.

This can easily be extended to the question of Kenku as such:

Most Storm Sorcery Sorcerers can speak Primordial, but if he/she's a Kenku, he/she cannot because of its languages.

...but you can speak only by using your Mimicry trait.

You would, however, now be able to understand Ignan, Aquan, and Terran and, after conversing with creatures who speak these languages enough, pick up enough phrases to use Mimicry for these languages too.

Why is the Kenku language more specific than the Storm Sorcery feature?

To identify which rule is more specific, you have to identify which rule is general. In this case, the general rule is Wind Speaker, because it can apply generally to all races while Kenku language restriction only applies to the Kenku.

If Kenku was the general rule, then Wind Speaker would have to clearly allow speech beyond the Kenku restriction. It is, however, clear that Kenku can not speak without Mimicry,

specific: (adj.) clearly defined or identified.

Similarly, the specific case of being mute would also restrict you from speaking with Wind Speaker (since it doesn't specify speech beyond normal means).