524s are covered at Community Tip - Fixing Error 524: A timeout occurred but I guess you already found that.
As the article states the error occurs only if your server takes too long, so you should never the it after just a few seconds. If you say you do, the only thing you can do is to note down the connection ID of that request and open a support ticket and have them clarify why the error would happen before the usual timeout.
As for #2 of what your host said, this is not a timeout from your webserver but - as stated - from Cloudflare when your server take too long, so you shouldn’t find timeout on your server but just long running requests.
Apart from the support ticket, the best thing might be to pause Cloudflare and check if you get any requests which take longer than aforementioned time. If you do, you will need to fix this on your server.
And no, changing to “Full” won’t make a difference (whoever said that was misinformed), your site is now just less secure.