The PS2 backwards compatibility is a good point. It's quite good, so there's not really much reason to have an actual PS1 anymore. The PS3 backwards compatibility is not very good though, it adds a good bit of lag, and can't output proper 240p, 480i is the minimum.
These other emulators will add at least one frame of lag as well, possibly several depending on the method of vsync. I've tried NES emulation on a Dreamcast, and I could notice the lag just by feel. Games I was familiar with, especially the original Super Mario Bros., felt noticeably off. Not as tight as the real thing. Isn't part of this about showing off the real console, and the authenticity of the real controller?
That's because the PS2 mostly does not emulate PS1. Every PS2 had the PS1 CPU in it as an I/O processor and some, many older models, had even more PS1 chips, so it was more or less native. All PS3's do 100% emulation back to PS1. Older PS3's have most of the PS2 chips for the same reason. The only PS1 games I played on my PS3 were RPG's (FF7, Star ocean 2, and others) worked great.
If the only PS1 game you wanted was Parappa the rapper a moded older PS3 should do fine. You could also dump backups of other PS1/2/3 games on the hard drive, if you don't want to have to mess with physical media.
I completely agree about only using the original hardware. I have over 20 original systems, mostly from thrift stores and yard sales. I just wanted to share ideas. If emulations 100% out don't get any of the new slim NES or the 3 in 1 systems. They all use system on a chip emulation. I would like to point out, not all emulations are equal. Try using an emulations made for accuracy like KEGA vs one made with lots of extra needless features.
Note:The special addition NES I posted to earlier runs games natively and they are much more reliable then the older classic gray rectangles.
Personally I would put in a SEGA Genesis/CD/32X on the 3rd CRT, They can play Master system and Game Gear games natively off a multi cart. Better yet I would put my collection of systems and games above the bar and use a switcher to pick witch system goes to the 3rd CRT or to the big screen.