I'm sort of a mix of both. I buy old hardware for the hardware, but I also chose based on what I need. As an example, the first 2 Thief games are best on a Window 9x machine with 3DFx hardware. Nvidia cards screw up the sky and ATI cards screw up the fog. There are fixes for them for modern machines, but the feel of the game changes.
On the other hand, I also have a TI-99/4A, a Commodore 128, an Amiga 2000, and an Amiga 4000. If I had the money, and I may never be able to get this, I'd also have an Apple II (preferably a IIGS, but a IIe would work if necessary.) Why do I have those? Because those are the systems I started with. My very first computer was the TI, followed by a C64 and an Apple IIe. The C64 eventually graduated to the 128, and I never had any of the compatibility problems I've read about. Finally I got the A2000 (after playing a lot on my fathers A500.) It was a long time before I gave the PC a try, my very first one was a state of the art 486DX50 (soon upgraded to 100.)
I love the old games, and many of them I'll play in DOSBox, but I also like playing them on compatible HW. But only if speed isn't going to be an issue. Privateer will be played on my real HW, while Wing Commander will be in DOSBox (as examples.) I'm still toying around with slow-down software for my real HW, but none of them are really satisfying. They work, after a fashion, but DOSBox seems to do it better.
As for monitors, I'd stick with my LCD if they would do what I want. I want to be able to set 4:3 and have it scale to that. Right now I have an HDTV being used as a monitor (it's getting switched back eventually,) and it won't even try. I cannot set it to maintain aspect ratio in any manner. Everything gets stretched. DOSBox is fine, as it displays at 1366x768 adding in the pillars itself. But my real machine gets stretched. The LCD monitor I'm putting back will maintain aspect ratio, but doesn't "adjust". So a 640x240 gets shown at 8:3 ratio, instead of the 4:3 the graphics are designed for. I really wish I could set the monitor so that it would display a fixed 4:3 when I tell it too. Or better yet, display everything except 1366x768 at 4:3. My main system never uses anything other than 1366x768 thanks to Microshaft and their "your to stupid to run your own OS," decisions (sorry pet soap box there.)