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In reduction of carboxylic acid with $\ce{LiAlH_4}$ the first step is deprotonation.
But in the mechanism of reduction of amide with $\ce{LiAlH_4}$ (here), the first step is addition of hydride on carbonyl group. Why doesn't the hydride ion abstract a proton from $\ce{-NH2}$? Since acid base reactions are faster than nucleophilic addition.
As compared to the carboxylic acid, deprotonation of the amide is negligible.
I don't recall any data for THF or 1,4-dioxane as the solvent, but F.G. Bordwell has determined a lot of $pK_a$ values in DMSO.
In DMSO, the $pK_a$ of acetic acid is 12.6 [1], while the $pK_a$ of acetamide is 25.5 [2].
In the mechanism on the page you referred to, a positive species ($\ce{HRC=NH2+}$) is formed, which is very unlikely in strongly basic conditions. I do believe the amide would deprotonate at some point, most likely in the first step, similarly to the carboxylic acid. Here is a link to a page which shows a more likely mechanism.
