No paint is going to hold up to chairs and doors banging into it, is it? I think the problem is that you are painting over a stained surface, so that when you whack it, that stain underneath is going to be visible again, which is really why it looks worse than if it was just raw wood underneath. You will need to wash off grease of course before priming, sand really well and use excellent materials. I don't know what else you can do except for tell everyone to be careful.

Research putting a layer of clear poly or lacquer like Acrylaq over the paint to make it more durable. I have never painted cabinets but I have heard of people doing this.

I haven't painted cabinets, but I've done a lot of research on GW about it because I am still mulling over painting my bath cabinets. What I've learned - prep is the most important (and least fun) part of the job. Clean them well, with a degreaser, then sand them to remove gloss. wipe down well to remove all the dust. Prime with a good primer, paint with a good quality paint. I've seen BM Cabinet Coat mentioned as a very hard finish. Some use oil based paints, but if you are going white or cream, the oil based can yellow over time, and I've also read that it can crack. But I have no personal experience with that so I can't say if it does. If and when I do paint my bath cabinets, I will do them as I've described.. But a kitchen takes more use and abuse than a bath cabinet, I think over time, even the best of paint will chip a bit. But, if you've done a good job, that will be lessened and you will have the extra paint for touch ups. Its a lot of work though. We refaced our cabinets and the prep work was definitely grueling and not fun. But, without good prep why bother, its not going to last unless you do it right.