A word that conveys that an act or expression or object had failed to fulfill its purpose

Solution 1:

What you're asking for is a way of saying that even though something is normally meant in a certain way, somebody is simply going through the motions in a kind of insincere fashion. They're not really concerned with the normal result of the action, but are doing it for the sake of appearances only.

An idiom that expressed this is for show:

[Merriam-Webster]
1 : intended to be seen but not used or bought
// We're not supposed to eat the fruit on the table. It's just for show.
2 disapproving : done in order to gain the approval or favor of others
// He says he enjoys classical music, but it's only for show.

An actual adjective that fits the example sentences is perfunctory:

[Merriam-Webster]
1 : characterized by routine or superficiality : MECHANICAL
// a perfunctory smile
2 : lacking in interest or enthusiasm

So:

The perfunctory donation was never meant to be made in good faith.
The perfunctory apology implies a lack of actual remorse on the directors of the board.