What is the logic behind not saying a bracha on counting the omer during the day?
The omer was not brought at night, presumably.
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The counting of the Omer is done at the time when the Omer was cut, which is done (ideally at least, see below) at night. It doesn't have to do with when the Omer was offered (Shulchan Aruch Harav OC 489:3)
As far as why we don't say a berachah when counting by day, he explains the reason is because there's a machlokes whether the Omer may be cut by day if it was not cut by night (which would reflect on us counting the Omer then), so because of the rule "Safek berachos lehakel" (if we're unsure whether to say a berachah or not, we rule to be lenient, and not say the berachah, in order to avoid saying Hashem's name in vain), we don't say a berachah.