Let’s is the English cohortative word, meaning “let us” in an exhortation of the group including the speaker to do something. Lets is the third person singular present tense form of the verb let meaning to permit or allow. In the questioner’s examples, the sentence means to say “Product (allows/permits you to) do something awesome”, so the form with lets is correct.
In "Let's get started", the starting point is in view and "Let's get going", you are on the starting point already. Moreover, there is a sense of extra involvement abundantly made clear by the sentence, " Let's start going".
phrases - Let's get started! or let's get going? - English Language ...
AHD let let (2) noun 1. Something that hinders; an obstacle: free to investigate without let or hindrance. It's also in Merriam-Webster: M-W let : something that impedes : obstruction ruled his little world without hindrance or let— "Let" as that noun has its own etymology: etymonline let The link says it's obsolete except in legal contexts.
As a rule, we don't use the passive voice with "let". "Allow" or "permit" is normally used instead: We were allowed to do whatever we wanted. We were permitted to drive the vehicle. Accordi...
Let go or be dragged. This is a saying often associated with Zen Buddhism (occasionally Stoicism). As far as I can tell, there's no historical connection, and it might well be originally English. ...
quotes - Origin of the saying "let go or be dragged" - English Language ...
Many people use "let, let's and lets" in conversation What's the difference between them?