The slider doesn’t disappear at mobile widths, but it does become too small to be useful.
Have you tried building that slider on another page as a standalone object using only relevant CSS, none of the surplus HTML and CSS that make the rest of your page work, to see if it works without the additional code in your site?
or You could make a copy of your home page and, using the copy, begin deleting irrelevant HTML and CSS links, testing the broken slider after each change until it no longer “disappears” at mobile widths.