I'm good with restoring the bakelite and drilling out the old knot. Please give me some advice on working with the black piece. Is it also bakelite or does it need some other treatment.
I have always treated the black part same as the bottom and I've had no issues. I can't swear that it's "Bakelite", but it seem to stand up well to light sanding and polishing. JMHE
I used wet 400 grit paper designed for auto body work and followed with (slow) bench buffer with jewler's rouge. If the scratches are so deep that I can't get them out with the 400, I just leave them and call it character. LOL I sanded pretty hard with a Dremmel sanding drum and accidently burned a lovely "Century" handle and had to toss it.
The black material on some handles are extremely soft and will not hold up well to aggressive buffing with a bench buffer. I have learned to use elbow grease instead, but bakelite really pops with blue compound on a loose cotton/flannel wheel. Drilling the knot out should be the same as any other brush, the black half is most likely hollow though so take it easy on the pressure to avoid damaging the base once you are through the glue.
It looks like the white stuff is in small grooves? The two similar handles I've done didn't have said grooves, so I can't be sure. But I would guess it would indeed come out with sanding. I usually start at 200 or 320 (unless I'm worried about the material, in which case I start much higher). I tend to go 200, 320, 400, 600, 800, 1000, 1500, 2000, 2500, 3000. Sometimes that just feels like too much work, so I just skip every other grit. Honestly I'm not sure it makes a difference. You could probably do 400, 800, 1500, 2000 and get the same results.
Don't quote me, but I'm pretty sure the black top portion is vulcanized rubber, which would be what makes it harder than plain rubber, but still softer than Bakelite. I've sanded the "rubber" portion without any issue. I didn't use a coarse grit sand paper, or at least not too long or with too much pressure. I may have started with 400, but 600 is probably a sufficient starting point. Good luck! I'm looking forward to seeing your finished handle.
Update: Just took a second look at your photo, and 400 might be a good starting point. Just go easy, and feel/look your way through it.
I started with the 400 on both the black and the red. The black part didn't seem happy about what I was doing...maybe too much pressure.. Looking good so far..
If it is just because black gunk is coming off as you sand, that is exactly what should be happening. Make sure you are using plenty of soapy water to clear out the debris that is lifted by the sandpaper. Too much pressure will only be a problem relative to losing depth on the lettering. The "scratches" are entirely natural and will be removed as you sequentially run through the grits.