After some more investigations I concluded that it's not possible to search by ticket type, SLA or service in the customer interface, in v3.1.10. If anyone finds out I was wrong please point it out.
There is this one configuration (Ticket::Frontend::CustomerTicketSearch###ExtendedSearchCondition) that I haven't been able to take under account because I didn't understood what it does (Help text: Allows extended search conditions in ticket search of the customer interface. With this feature you can search w. g. with this kind of conditions like "(key1&&key2)" or "(key1||key2)".) I saw other people wondering out loud quite recently about it but no answers.
Finally, if anyone else is wondering, even though you can't enable your customers to filter tickets in their searches by ticket type, SLA or service, you can still get those fields included in the output produced for CSV by adding to the Ticket::Frontend::CustomerTicketSearch###SearchCSVData configuration the values: Type, SLA and Service. This is really useful for customers who need to compute their own reports, something that can't really be done via the customer interface once you go above a couple tens of tickets due to the interface lacking filters and customizable grid columns.