Too many people think that just saying, “I’m sorry.” means the end of it, that the person being apologized to should take the apology and be done with it. That an apology should come with automatic forgiveness.
“What, I’ve apologized for that, why are you still angry?”
To me, an apology needs to contain three things. An acknowledgement of what you did wrong, an understanding of how what you did wrong harmed the other, and a sincere commitment to avoid that behavior in the future.
It is then up to the aggrieved party to determine if you are correct in what it was that you did wrong, whether you understand how it hurt them, and if they trust your promise to not do it again.
If they do not accept your apology, it is because they do not feel that you have covered this, they do not trust you to not harm them again in the same or similar way. Demanding that they forgive you, when you have lost their trust, is only going to cause anger and frustration.
If you are using an apology as, as the OP put it, “Getting things over with sooner”, then you should not be too surprised when the person that you have harmed is not agreeable to your timetable, that they do not accept that expediency on their part is the priority of getting over being wronged.
If you don’t think that you were in the wrong, but the person that is harmed does feel that you were in the wrong, making an insincere apology, where you do not acknowledge what you did wrong, how that hurt them, and express a commitment to avoid doing so in the future, then they are absolute in the right to not accept it, and in fact, be angry at you for trying to “smooth things over”, without actually addressing what the things are that you are refusing to actually confront.
Many people then blame the aggrieved party for being unreasonable and not extending forgiveness to someone magnanimous enough to say, “I’m sorry.” even when they aren’t. It’s not exactly gaslighting, but it’s definitely on that end of the spectrum when it comes to abusive relationships.