There's no implied requirement for any given user to be proficient with an item in order to attune to it

Borrowing from this answer to a similar (but unspecific) question, the key take-away I've made is that while Magic Items are fully allowed to specify restrictions on who can attune + use them (which can themselves require the user to be proficient), if it doesn't say it requires proficiency, then it doesn't.

I think this stands to reason given the prevalence of magic items that don't have proficiency doled out under normal circumstances. Items like bracers, cloaks, and wands—the latter of which I think is the most interesting in this context—don't offer the ability for wielders to be proficient (for, arguably, obvious reasons), yet so long as the user fulfils any prerequisites the item does demand, they'll be able to use it.

Caveat: How does the effect work?

If an item requires the user to do something, either for attunement or use of the attuned item, where that something represents something that cannot be done without proficiency in the item, then that implies a requirement of proficiency. Considering the following item I've made up on the spot:

Magical Platemail of Effervescence

(Requires Attunement) Whenever the attuned casts a spell while wearing this armor, all affected targets begin to emit bubbles from their nostrils

A character cannot cast spells while wearing heavy armor unless they have proficiency in heavy armor. Therefore, proficiency in heavy armor might not be a directly stated requirement of attunement for this item, but the user would be unable to invoke its effect without being proficient in heavy armor.

In your specific case, the Wizard can probably use the sword just fine

What the wizard needs to do in order to invoke its effect is merely draw the sword—an action which in-and-of-itself does not require proficiency. Proficiency is only required to make the wizard competent at attacking with the sword, not to merely hold it or draw it from a sheathe.

You didn't specify conditions beyond that, so there could be tricky parts to the sequencing: does the effect go away if they put the sword away? Do they have a hand free to perform the somatic/material components of the spell they want to gain the effect on? But as long as those considerations are met, then the wizard would be perfectly capable of gaining the benefit of the sword, even without proficiency.

No, because that would be to "function" as a magic weapon, which it explicitly lacks the ability to do

The key (as I see it) is parsing the word "function."

Generally, the function of something is its capacity to act or behave in a certain manner, or to have a certain purpose. So an item's ability to be used in combat so as to overcome any specific resistance is certainly a "function" in the ordinary sense of that word.

Thus whenever the sword "functions as a normal (not magic) blade" then it lacks the functions that are associated specifically with magic ones, including the function you are asking about.