Sure cutting the dash looks crappy, if it's done poorly.
On the other hand, if you choose a head unit that looks good, and have a professional stereo shop handle the installation if you don't feel comfortable wielding a sawzall, dremel or whatever, then it can look very nice.
If you swear you'll return your unit to stock someday, you can always mount your modern cd player under the dash, in a plastic sleeve. Like I said, this worked great in my '66 Fairlane--it was mounted much in the same way as many people mounted 8-tracks or FM receivers back in the 60s.
Lots of head units have little words printed all over them advertising the technology they're compatible with, and I think these look awful. Pioneer's head units don't, which is why I like them, compared to, say, a Kenwood unit. If you were to just do a little research, you could find out that Pioneer makes a great quality unit called the DEH-P880PRS that even has the classic two-knob interface.
http://sonicelectronix.com/item_6822_Pioneer+Premier+DEH-P880PRS.html
check it out, I think it would look great in a classic Mustang, and know a few VMFers already are using it.
This is the one I plan on using:
http://sonicelectronix.com/item_9008_Pioneer+Premier+DEH-P690UB.html
mainly because of it's USB-linked Ipod interface, which allows you to scroll through the songs using the rotary knob at the same speed you'd scroll through on your ipod.
Both of these are great looking units, and they're plenty of others out there, too. If I could afford it, I'd put a McIntosh receiver in there, which is the same kind that Ford used in the new GT. If you don't mind dropping about $1600 on a head unit and amplifier (they don't have an internal amplifier for clarity reason) but want a CD player that looks like it could have been designed in the 60s, McIntosh is the company that makes it. Extremely high-end stuff.
I think any of these options look and function better than having to open your glove box to change the radio station or even the volume.
So far as security goes, nearly any modern receiver has a removable faceplate which can be locked in your glove box, trunk, wherever. The receiver is useless without the faceplate to control it.
I'm certainly not trying to tell you how to build your car; it doesn't matter to me that much at all. The original poster just asked for advice, so that's what I'm giving. There are options out there other than hiding your cd player away, and good looking options at that. Finally, do what you want, as you will anyway. It's your car.
For me, I'll have a modern CD player mounted in the dash of mine.
-Joseph K.