In this case the alt name on those segments should be as you described S-County Code-Road Number. I can lower those segments for you to correct that, and add the correct city name as well. The designation of highway or not comes from DOT, just not these numbers, from something called Functional Classification (http://info2.scdot.org/sites/GIS/SitePages/FunctionalClassSearch.aspx) we have a wiki for that too (https://wazeopedia.waze.com/wiki/USA/Road_types). Since you are already in Discord it may be easier to discuss over there.
Also the S-XX-XXX is not very useful for anything except it does help us find closures easier since SCDOT sometimes only gives the road number and not a name. We decided we would map roads with the correct alternate name for this reason mainly, but its not a huge priority for us either.
I personally would not spend too much time working on these, so unless you really want to there are other things we have that you can work on like places. There’s value to marking S-roads as jwe mentioned, but it’s probably the state’s lowest priority.
Also, if you do read the wiki, I want to point out that the S-numbers do NOT indicate that the roads are county highways. We do not have county highways in the same way that some other states do. Around half of SC’s total roadway miles of all types are S-number roads, and marking all of those as PS would prevent Waze from organizing roads properly for routing and clutter display.