I have a Classic 350 with only about 600 miles on it so far and haven't yet ridden with a pillion, but in the early 70's I had a 250 Crusader and that was fine with a pillion on the sort of roads on which one would ride such a thing. However, it would only do about 50 miles between big-end rebuilds
The 350 J series engine does appear to be a nice unit, as is the gearbox. I would add these observations:-
As supplied, it appears that almost all Classic 350's have the gear shift lever set too low - unless you habitually ride in sandals and not boots. To adjust is simple but does require moving the frame slightly as the lever fouls the footrest support and won't come off.
Said footrests are essentially horizontal and, even with the shift lever raised a spline, it is still difficult (for me at least) to change up wearing boots. This has been solved by fitting a heel and toe shift (contrast with the Meteor comment above) - £28 posted quickly from India and appears to be a genuine RE part, fits very well and shifting is now a breeze.
Third, and most important, is that the first service is at 300 miles and, according to RE, should include a check / adjust of the valves. I quote from the Service Manual - "Valve clearance setting at First Service (500 km) and above mentioned internal service period instruction is mandatory".
My own experience, and that of many others, is that a lot of dealers DO NOT undertake this check at the first service. I knew for sure that they didn't do mine and I didn't complain because I wanted to do them myself anyway. I did them the day after the "service" and found that the inlet valve had ZERO clearance (should be 0.08mm iirc ) and the exhaust had about half the 0.18mm iirc that it should be.
Checking and adjusting the valves is a simple job but is quite time consuming, as you have to remove the tank and the rocker cover gasket is a bit of a pig to get back in.
I've only ridden in cold (and wet) weather so far and am really looking forward to some warm weather riding around mid Wales on the little Enfield.