In general, someone who misses a complete day's counting the omer continues to count without preceding his count with the b'racha (benediction) normally recited for the mitzva of counting. (In fact whether he should recite it is unclear, and we rule that, because of that inclarity, he should not.) I'm wondering about a shatz (leader of the prayer service). In many synagogues, he recites the b'racha aloud (also having in mind to exempt anyone listening). What if the shatz has missed a day? Presumably, like anyone who missed a day, he doesn't recite the b'racha aloud — but is there an exception made for a shatz, whether so as not to embarrass him or for some other reason?
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If the Rabbi of a congregation forgot to count one day and he usually makes the bracha out loud, he may continue to count with a bracha; by not continuing to count he will cause a disgrace for Torah and is a disgrace for the members of the congregation.
(Shevet Ha'Levi 3:96, 4:157 note to ch 96)
The heter is for a Rabbi because of his public position, not for every Chazzan.
Despite the regular principle of "Safek brachot l'hakel", when in doubt do without a bracha, the case of sefirat ha'omer may be different. This because only the Behag understands that it is problematic to count if one lost count. Because the opinion is remote, in a case of embarrassment we could rely on the other opinions and allow the Rabbi or Chazzan in this situation to make the bracha.
Josh Waxman, author of The Parsha blog makes this argument here, citing a second hand account that this was the opinion of R' Vozhner, and Rabbi Ari Enkin agrees with the conclusion in the comments.
Can't bring you sources, but I've seen it happen various times over the years that the Chazzan asks somebody else to recite the Bracha in his stead.
This can easily happen to a new mourner who is exempt from evening prayers and may miss a day, as he is distracted by his misfortune. (The Kitzur Shulchan Aruch in 196:18 says that he should count after the burial. If that would entail missing a day, he should count without a Bracha so as not to miss a day.)
In certain shuls (like by the Yekkes) the Rov always counts the Omer first; possibly for this reason - that he's least likely to have missed a day. We also gain that we don't have to worry about embarrassing the Shat"z nor worry about him making a mistake as to which it is.