Can employee stocks affect market value?

Sorry if this is the wrong spot for this post, I’ll post it elsewhere if needed

Just made out great on my first batch of an employee stock program. I believe it was about the standard that most companies provide:

Take up to 10% of each paycheck over 6 month period
Receive stock at the end for lower price: first day of 6 month period or last day of 6 month period + 15% discount

As far as taxes go, I believe you are said to have owned the stocks since the beginning of the 6 month period, so you must hold another 6 months for tax break

So I received my stocks a month ago, our market price went up about 10-15% shortly after receiving, and then dropped after the earnings call a few weeks later. I was lucky and sold almost immediately and made around $1000 because I didn’t think the earnings call would be very good. Now I’m back in with 50% more shares than before.

But it got me thinking about how this works.
Do these shares come from a company “stash” of them? If there was a mass purchase straight from the market it would drive the price up at the start/end of these ESPP periods right? The price went up right when we got them but the CEO was making some moves so I didn’t think it was the employees getting their shares changing that. I would think a lot of people sell their shares very soon after receiving them, and wouldn’t a massive sale drive the price down as well? I mean it’s a big company but I also am probably really putting a lot of stake in how we could even change anything with our shares.

Also, does the company benefit from this in anyway? I was being extra conspiracy driven for a bit, when it seemed like the price suddenly dropped back to the same price at the beginning of this program, I don’t really think they would benefit with us all getting shares at a better price but I’m sure the employee stock program has a lot more benefits for the company than the employees and I’m just curious what that approach may be.

I don’t know if this varies too much between companies or if it’s just a dumb vague question, I’ve never worked for a large company so I’ve never had to consider any of these things.