North South
Sorry to point out to you that you are wrong regarding PS. You pass traffic information on conflicting non-participating traffic to PS when necessary - not not-at-all as you suggest. Perhaps you are confused because you are not responsible for applying procedural separation to non-participating traffic, which of course is a different matter.
You say radar can't be used with PS. As such, you imply you would not pass radar derived traffic information to PS. Not sure that would be a reasonable action if you considered there was a definite risk of collision. For instance, think about unknown traffic attacking your traffic in a Procedural Hold in Class G. There may be very little you can do to deconflict the traffic but are you seriously suggesting you would not call it from your radar picture even if you possessed the skill?
"Traffic believed to be you has traffic....." will be defunct as BS and PS requires the passing of generic traffic information on definite collision risks - the same with PS. This is because controllers should not to imply the use of radar when passing such information. However, I am with you when it comes to using the phrase in essential circumstances (i.e collision situations). I say, if you have the skill use it. After all, I need to sleep at nights.....