I too find the phaser sound quite satisfying, and also wonder sometimes why that is. I think partly, the added low frequency gives it a feeling of depth. I used the phase-shifted sound on my 1977 album Dr. Banjo Steps Out, and it got a lot of surprised and mostly good reactions. I used it throughout the Hot Rize years, but kind of sparingly, as I felt it fit only a limited number of songs (including two of our "hits", Shady Grove and High on a Mountain). I still use it occasionally when it seems to fit.

No need for a pickup, but a good mic with good low end response will help. To hear it, you need to play it into headphones that block the actual banjo sound, which would make it hard to hear the phased banjo sound. It can be recorded, too, and then nothing but the phased sound can be heard. I use a phaser I got in 1974, which is still made as a reissue (MXR Phase 90) but not with the same sound, unfortunately. I have been able to imitate the sound on some other devices, but it takes some careful tweaking.

In other words, I would suggest you try what's now on the market and see what you like. You could go in a music store where they sell such gear, bring a good microphone and good headphones and set up to play into a good amp. You can turn the amp volume down and use headphones for monitoring. Often effects like phasing (also called flanging, and similar to "chorusing") are combined into units with many effects. But the single-effect boxes are much cheaper. You don't need to play anything fancy, just a few chords and a little picking, just whatever you play.

Pete