your Becker Mode S transponder has an extended squitter, why not run an FTDI cable from one of your Rosetta USB sockets to the transponder and have ADSB Out?https://uk.rs-online.com/web/p/kvm-mixed-cable-assemblies/6877828/
(a)Only ATC can guarantee to see your Mode S by interrogating it and receiving a reply i.e. doing what it was designed for. ATC don't share that information with the likes of FR24.
(b)FR24 (and similar) can only see your Mode S by triangulating your position (multilateration) once you're seen by 3 or more ground stations. These are generally 'volunteer' stations, so there's no guarantee of their location and they're not necessarily going to see you at low level or when you're hidden by hills etc.
(c)Sometimes FR24 need your specific aircraft in their database to show you on their website. This isn't always the case and does seem to be random. If you want to better your chances then register your aircraft with them.
(d)You stand a FAR better chance of being seen by FR24 etc if you have ADSB Out, which is easy to do if your transponder has an extended squitter.
(e)PAW is not seen by FR24.