Or perhaps you had airliners which were flying non-RNAV sids/stars using some old FMS which used VOR/DME as the nav source?
Nope, not as long as you don't count RJ85s, ATRs and such as old.
Our "old" stars where conventional with RNAV overlay, and even on straight sections of stars some customers often wandered off centerline 2-3nm, and when asked said they where "on magenta line". Back then we did not focus much on the RNAV status on the aircraft, but surprise-surprise when we went PRNAV last year, the handling of BRNAV flights singled out some customers, and the "offenders" was all among them.
You are right that many BRNAV a/c are fully capable of flying very accurately, which is nice for us, but as long as some don't, we can't count on it.
And that brings us to what GWYN asks about.
On BRNAV procedures experience shows that a/c often overshoots turns with several miles, even on fly via points, not all the time, but several times each day. On PRNAV procedures my experience is that a/c nail the turns rolling out on the next segment each time.
(When it happens nowadays, the crew mostly confess to having gone into heading mode or some such, and not trusting the box. Or cancelling the procedure in the FMS, and wandering off whilst trying to put it back in, without telling ATC
)
When i have a SID that turns parallell to a STAR with say 7nm, any overshoot on that turn will mean a loss of separation, so with BRNAV you would never design them that close, or you would use vertival separation until the turn was observed to be accurate, or
turn the departure onto the desired heading earlier to assure separation. Hence increase worload.
With PRNAV procedures that are properly designed, you can IMHO shift more aircraft in a given volume of airspace with the same staff and ATM equipment.
But: For GA IFR you will as peter correctly states seldom fly a SiD or STAR, and the issue is speed. Modern terminal procedures are designed to manage a even flow of aircraft, i.e capable of flying arrivals at speeds in the low 200s in clean config. Departures will often see the use of "prop" SIDs for turboprops, since performance is more diverse in climb.
In your GA hotship tourer you will often be the slow boy, even for prop departures, and ATC want to move you away from the flow.