I would say that if you plan on racing, a modern bike is a lot more convenient for changing set up and all that. But if you just want to ride the track and do some training for fun, an NJS bike is absolutely fine, and of course there are aesthetic considerations. Who doesn't think an old-school steel track bike is more beautiful?
As far as flats go, they are so rare (for me) on training wheels that it doesn't really concern me. The single time in over 10 years of doing this that I flatted a training tire, a friend simply lent me a wheel. Also, on steeper tracks, in the event of a flat while riding I'd be a lot more confident in a tubular keeping some rubber between the rim and a track surface, possibly saving some skin.