Shouldn’t we use "lots of" with plural nouns and "a lot of" with singular ones?

The source that the questioner links actually says

There's a lot of aspects of being in power that seem "unfair."

The writer has apparently chosen the singular 's (is) to agree with lot. That makes a certain amount of sense, but “a lot of” followed by a countable noun in the plural has come to be most often used as if it were many, and we would say “There are many aspects . . .”—so likewise most English speakers and writers use “There are a lot of . . . .” Note how this NGram shows “there are a lot of” as more frequent than “there is a lot of” even though many instances of the latter must be with non-count nouns, as in “there is a lot of resentment on both sides.”

P.S.: In response to revision of question, I have revised my NGram link to show “there are lots of” running third in popularity.

A lot of things means a large number (plural) of things, hence you will use a plural noun (unless the noun is uncountable, then singular). A lot of apples, a lot of chairs, a lot of questions, (but a lot of water, a lot of sand).

Lots is just the plural form of lot. And so more than one set of a large number of things is, well, still a large number of things (and thereby plural), so you still need a plural noun (same exception for uncountables). Lots of apples, lots of chairs, lots of questions, (but lots of water, lots of sand).

They will both generally agree in the plural, although when the focus on a group as a single unit, the singular agreement is possible.