A good storage 'scope will catch events you'd not be able to see on an analog 'scope - no amount of upping the brightness will persist a 50ns event for long enough to be see by the human eye.
Having said that there are storage scopes using a storage tube that do solve that problem without an ADC: History of the oscilloscope - Wikipedia - find a secondhand one?!
I've just moved from analog to digital scope - old analog one I built from a kit / electronics magazine design a long long time ago, too slow really. New digital scope is nice by comparison 