They reduce sharecount. Not all buybacks increase share price. I can give you a thousand examples of massive buybacks that happened before the stock dropped considerably. But all reduce sharecount.
Searching on the subject of "buybacks where share price decresed", Google returns Merck as an example (reffing Harvard Business Review), which actually works quite well to illustrate. The HBR article even notes the main strategy "historically, companies that bought back their own shares have posted immediate returns between two and 12 percentage points above the market average"
"Merck's stock price dropped after a major buyback announcement when investors focused on expiring patents and a drying drug pipeline"
Except: Feb. 23, 2000, NYT, "Merck & Company, the No. 1 United States drugmaker, will buy back as much as $10 billion of its shares, which have fallen 18 percent this month." (Closest share price I can grab is 2/25/2000 at $57.39)
Share price then climbs steadily (tiny drop in July) up to a max at 12/29/2000 of $89.27 before finally crashing.
The first example Google returns is full of info on the stock behavior that makes it look like the stock buyback did not initially jack the price. Owners had 10 months to pull in a 55% share price increase before it crashed. And they floated through the 2000 March 10 bubble popping until the stock market really started deflating in 2001.
Make up whatever nonsense you want about the “fundamental purpose” of something, it doesn’t matter. The purpose of a system is what it does:
https://en.m.wikipedia.org/wiki/The_purpose_of_a_system_is_w...
Stock buybacks increase share price. There’s no reason to look any farther than that. The purpose of stock buybacks is what stock buybacks do.