Don't worry about it hardening, that's just the body's way of encapsulating it so it will ultimately go away. You removing it is not what should be done. Here's a reference about the topic, it's very informative:
https://www.medicalnewstoday.com/articles/312361#how-to-remove
One of the interesting parts of the writeup is that if "part of the sac wall remains on the skin" the cyst might form again. That seems to explain why your home attempts to remove it were not effective in the long run.
I had one of these in an earlobe once, just a hard little lump. I never tried to do anything to it, because it didn't look funny or bother me, but it was there. Then one day months after I noticed it, I idly felt my earlobe and the cyst was gone. Probably my body simply absorbed it. If your cyst bothers you a lot, you could ask the doc to remove it, but it might take care of itself over time.
I've had a couple of these in my life. I never was given any medication for it, don't really understand that. Once a doctor just flicked the whole thing off and walked away. That was the end of that one. Once I was just told to soak it with a washrag soaked in hot water repeatedly and that took care of that one. When you do get rid of it, try to keep the area cleaned, this is a sweat cyst basically, but it's nothing to worry about and they do go away. Peace.