I AM talking about the underlying motivations and they do impact...

I AM talking about the underlying motivations and they do impact market dynamics. In general: more people hitting the bid price lead to decreased priced and more people hitting ask prices lead to increasing prices - it's as easy as it is.

Imagine a model (where shorting is banned) with two people and two shares. Person 'Retailo' is a hardcore long holder, he holds one share and for the sake of the model he would buy another share at bid price.
Person 'Alfonds' also holds one share, he would sell for the ask price, but since nobody wants to buy for that price there is no transaction and prices stay the same.
(both might change their behavior in the future or other market participants may enter at a later date, but that doesn't matter for our thought experiment right now)

Imagine a different model with three people and two shares.... thinking about it, let's make it four people! And short selling is allowed! Setup is the same as above. Additionally to 'Retailo' and 'Alfonds' there are two other players: 'Shorter' and 'Shorterslittlehelper'.
'Alfonds' decides to lend his share to 'Shorter' because better than selling to Retailo (with his boring bid offer) he would rather wait for a later time when prices have improved. By lending to 'Shorter' he makes money in the meantime and he can always ask the share to be returned when he changes his mind and wants to sell for a lower price.
Shorter has a different opinion of a good selling price than Alfonds. He is glad that short selling is allowed - otherwise he could not do what he is doing now because.... he initially has no share to sell! So he borrows the share from Alfonds and dumps it on the market. Now, depending on who placed the bid order first and/or at what price, Retailo or Shorterslittlehelper get that share. One can imagine that c.p. the buy side is weaker now: Retailo's bid might be gone because he got the share (Shorterslittlehelper might even pull his bid now to make the buy side even weaker) OR Shorterslittlehelper bought the share at bid price and Retail's bid is still in the market. In the latter case, Shorterslittlehelper could now just lend the newly bought share to Shorter. Shorter sells to Retailo (and knocks out his bid) and is now two shares short (he borrowed from Alfonds and from Shorterslittlehelper). Funny thing is: it's the same share!
And oh wait, Retailo got two shares (one he initially owned and one from Shorter) but they are potentially "worth less" because market price has been lowered (or the spread widened)? Are there even buyers anymore?!

[One has to expand the model here with more SOI, more repetitions of selling between Shorter & Shorterslittlehelper or introduce other market dynamics like stop loss orders or margin loans, maybe even some FUD to convince Retailo that he should sell at a lower price.... but you get the point.]

tl;dr:
short selling banned = no change to bid/ask in the model
short selling allowed = buy side is weakened, the "price you see" (last price) is lower (former bid price)

The model ends with a sizable short position for Shorter. Do it 613m times and you get the situation we are now in.