Since your reply in 2019 to this community question, Jira Service Desk (now Jira Service Management) has changed the interface. It was a bit convoluted to find how to do this in JSD/JSM (Cloud). That said, my question is why the search box in the upper right not do the text search properly in a similar way even if I use the search syntax such as
comment ~ "some text"
Today the upper right search box (the one with the magnify glass) using the above syntax doesn't give me any result at all, but if I input "some text" into that search box I get 1 hit, while if I use the "Advance issue search" from the "Filters" dropdown I get 3 hits. The search box also doesn't appear to search in internal comments.