To me the expression
Let's simply share
I've always thought the adverb should come after the verb. Is that correct?
"Let's simply share" and "Let's share simply" have different meanings.
'Let's simply share" has the same meaning as "Let's just share". That is, let's share without any more fuss, or making too much of sharing.
"Let's share simply" has the meaning of sharing in a simple way, that is, to avoid complication which could arise if we were to attempt sharing like "You can have 20% of the cake, and I'll take 35.47%, and the rest will go to Jake." That's bound to bring some sort of trouble.
They have different meanings because the adverb has changed positions. Adverbs can go anywhere. For example:
Each of them means different things, and both of them are grammatically correct.