I have no access to short term or long term disability through my job (I work for the Government), and have a position where I'm probably more likely than baseline, but not highly likely to come up disabled in the short term. I am also the sole wage earner in our family at the moment.
So, I bought disability insurance. I actually chose to buy that over life insurance, sticking with the $450k policy I get through work. I have enough sick time to cover a middle of the road illness, but not enough to cover something greater than about 6 weeks. My options are a bit limited given my work, but I was able to get a policy that has both short and long term rolled in together, and the long-term is 4 years 'own occupation' (which, since my job is Government-only is easy for me to qualify for) and 'any occupation' after that.
If I were in your circumstances, I likely wouldn't have bothered. In my case, we're a one-income family with three kids. We're still a good way away from FIRE, and don't yet have the huge pool of assets and passive income that a lot of people here have. Plus, being unable to work for any significant time would probably mean a relocation for us to somewhere with better health care, which would strain our resources... so the income would be needed. It's $40 a month, and the peace of mind we get is worth it. Once we have stronger financial resources and side-hustles, I'll re-evaluate.