Here it does meet the condition of pronunciation. The accent makes it so that it’s pronounced al.kov, not al.kɔv. Without it, the word would be pronounced with an open ‘o’.

The TLFi mentions:

Rem. Fér. 1768 et Fér. Crit. t. 1 1787, s.v. alcove (sans accent circonflexe), font observer que la 2esyllabe est brève. La durée longue est signalée à partir de Land. 1834. Littré : ,,Le dictionnaire grammatical de 1784 dit que l’o est bref : al-ko-ve. Aujourd’hui il est long.“

My translation

note: J-F Feraud’s "Dictionnaire grammatical de la langue françoise" of 1768 and "Dictionnaire critique de la langue française" of 1787, under the heading alcove (with no circumflex) observes that the second syllable is short. The length is indicated starting from Napoléon Landais’s "Dictionnaire général et grammatical des dictionnaires français" of 1834. Littré [1873-1883’s dictionary]: "the dictionnaire grammatical of 1784 says the o is short. Today it is long".

So the word was pronounced /al.kɔ.və/ initially and until the start of the 19th century, when a variant with a long (and tense) vowel /al.kɔː.və/ > /al.koːv/ gained popularity. Such sporadic lengthenings aren’t uncommon, especially with learned or fancy words, like technical architecture terms such as alcove. To reflect this change of pronunciation, dictionaries started using an accented spelling alcôve, which prevailed.

(Anecdotally, I suspect this late lengthening was regionally limited, since I’ve almost always heard it with an open vowel in Belgium. Can someone comment on the pronunciation in North America?)