I think it's like this... the A1200 outputs native screenmodes ranging from 15kHz through to 31kHz plus.
VGA monitors' lowest scanrate is 30KHz-31KHz (seems to vary a bit), therefore any 15KHz modes will not work with them without a scandoubler that doubles this rate to 30KHz+. Optionally a "flickerfixer" will cure interlace flicker too.
Old games and demos that bang the hardware, i.e. are not "retargetable" will usually output a 15KHz signal so will not work without a SS. Games/demos that allow you to choose screenmode will work if you select a 30KHz+ one.
However, some A1200 modes (e.g. Multiscan Productivity) are output at above 31KHz anyway, so these should work. It depends what you want to do.
There were a fair few "dual sync" monitors that scanned in both ranges, CBM 1940/42/50/60, Microvitec 1438/1538/GPM1701 etc. Have a look
here:
Amiga MonitorsI use a Microvitec 1438 on my A1200 and a GPM1701 on my PC, and because the 15kHz modes are video friendly, if I got a new PC monitor the 1701 would go to my PS2.