Paying dividends to shareholders accomplishes the same thing as buying back stocks: both return capital to investors.

Increasing the company pension would not have saved them from their current woes: they would still need to issue debt or equity.

Not exactly, because share buybacks are optional. A shareholder can decline to participate, increasing their stake in the company (meaning the investor believes that the net present value of the shares is higher than the price offered under the buyback).

It’s not common, though (and for the purposes of the comment you responded to, the distinction isn’t relevant). If management are saying “we can’t think of anything to do with this money”, most investors won’t disagree with them.

A shareholder can also reinvest dividends. Doesn't that accomplish the same thing as not participating in a buyback?

Only if the company allows them to (or is simultaneously raising capital). Also I believe (though not 100% sure, depending on where you are) that income tax is paid on dividends, irrespective of whether you subsequently reinvest.