I am always in a similar situation in Valencia (Spain) when trying to book appointments. Fortunately, there is an integrated appointment system here and it's fairly trivial to automate checking for open appointments using their API. I can usually get one in a few hours by checking once per minute for free slots.
I also wrote something similar for getting a NIE appointment (foreigner's ID card) by using puppeteer (headless Chrome for node) to actually fill out the website for me, about once per 3 minutes (max without getting rate-limited).
I'm fortunate to have the skills to do so but I feel bad for the rest of the people who have to check websites for weeks at a time to get appointments.
I also wrote something similar for getting a NIE appointment (foreigner's ID card) by using puppeteer (headless Chrome for node) to actually fill out the website for me, about once per 3 minutes (max without getting rate-limited).
I'm fortunate to have the skills to do so but I feel bad for the rest of the people who have to check websites for weeks at a time to get appointments.