minhag - Does one need to fulfill the 248 words when reciting Shema outside the timeframe of Kri'at Shema?

O.C. 61:3 states that one should recite a total of 248 words when saying Kri'at Shema. As there are only 245 words, the last 3 words are fulfilled by listening to the Shat'z repeat the ה אלקיכם אמת. Rama"h in says that there is a minhag that a yachid (individual) should say the 3 words אל מלך נאמן as a substitute.

I am unsure if this requirement is directly attached to the mitzvah of reciting Kri'at Shema, meaning during its timeframe. If I understand correctly, one who recites Shema before or after that period fulfills the mitzva of Torah learning, but not of reciting Shema.

So, does one who prays individually need to recite אל מלך נאמן. And, perhaps, more practically, many shuls say Shema after the time on Shabbat and Yom Tov mornings and before the time on summer Friday nights. Does the chazzan need to repeat ה אלקיכם אמת or does this accomplish nothing?