@SoAn Supervisors tend to do that even with those who do not have mental illnesses but are not moving as fast as they would like. I was in my early 40s when I graduated with a BS. I also wrote an undergrad honors thesis and later published it in a peer-reviewed journal. Later, I had a falling out with that professor, so no LOR for grad school, despite my graduating Summa Cum Laude. It was largely due to my mental illness, but also the professor's mental illness, as he admitted.

The alternative option that many students take is finding a research mentor at another school or institution when you are ready, and then asking if you could be a co-author on a paper. Then do that again with a second person at the same or different institution. Once you get two papers published as a co-author, and two letters of recommendation, then you can apply to PhD programs. Otherwise, you can do a Master's and try again with research while there, and maybe getting that published as first author while being co-author on other papers. You can rebuild your recommendation pool that way, too.

You are young. You have time to heal, learn to manage your depression better, and prepare before going to grad school.

Many students seek those opportunities after graduating as an undergrad. It is how you use your time to prepare that counts. You can do it.