I'm a bit hazy on those adaptors, they need a monitor that can sync down to 15Hz horizontal frequencies which is rare these days, but I think the kicker with them is that they don't allow display of PAL or NTSC through, so basically all the games will output something unusable since they force their output into these resolutions (as they normally write directly to the hardware and bypass all this fancy operating stuff

).
For me I used something like the scandoubler they mention in the link
Although that only works with the AGA chipset which means you need a A1200 or A4000, a little bit fiddly since you need to clip things over chips, but they work pretty smoothly, the A1200 even has a spare output port you can use for it. You will still need a monitor that can sync to various frequencies though, something that can handle say 30 to 60Hz horizontal frequencies should be fine, the old NEC multisyncs used to work a treat. The reason for the syncing is that these doublers just double whatever you feed them, so you get some odd frequencies out (and doubling DoublePAL isn't advisable! they do get hot

Although it does work)
There was an external version of it in a clear plastic case, the advantage of course there is that it works with the A500, the downside when I had it was that it overheated a little so I'd get lines on screen after a while especially doubling something that was already > PAL or NTSC

Could add a fan to it ideally.
One other option for display, if you have a big TV then hunt down a RGB scart cable (or make one up), it has to be RGB scart not composite scart, but that works well for games, my old Toshiba 32MWDB7 32" widescreen has one of its three sockets marked as RGB scart (its about 10 years old though!).
Your last option is getting hold of a pucka frame rate converter, serious money then though at £200+ I guess for a basic unit.
The easiest option really is just to emulate things, especially if the titles you want to play are all old A500 games early in the Amiga's life since the emulation is really good now, but if you want to play with the hardware then I'd recommend getting either a A4000 or a A1200 with a hard drive and build up your collection of hard drive installed games.
If your interested, that A4000 I'm using has an internal scan doubler which works well, actually it has a 060/50Mhz, ultra wide SCSI and a 10k RPM hard disk and a CybervisionPPC graphics card (basically a diamond stealth S3 graphics chip which works wonders in something that old

) the SCSI is a bit of an overkill, but I came to the conclusion that drive made no difference in the PC performance years ago.
And the 060/50Mhz card flies, the Amiga's operating system is small and very efficient, so a processor as powerful as this just gives raw speed, I've seriously never felt an upgrade with the PC that was as big since. There are all kinds of setups around, some using the voodoo cards via custom hardware etc, but I'd only go for that if you really want to get involved using the proper hardware.
I hope that helps.