Wish I had noticed your thread earlier, as I've installed butcher block counters in kitchens a few times...so if the info is too late, it's too late.
A belt sander and orbital sander are your friends. A belt sander is the best way to radius thick stock safely unless the budget is big..doing it with a router will be a little tough, but since you don't have to pay yourself, just go slow and careful.
From what I've read you are getting good advice about removing most of the cut with a saw and finishing with your router template, but that spacer on the bottom of your router is a bit concerning.. I would make a base that sits on the piece, rather than the tool.
Probably at least 4 times the size of your router base diameter...if the base is 6 inches, make a 12 inch by 12 inch base, like a chunk of 2x12 or maybe one of the other shelves, and clamp it to the work piece securely, maybe under your radius template, with the radius rough cut out of it at the bandsaw, like the shelves before they get routered.
As mentioned, that long router bit that's almost two inches higher than the work piece will probably deflect a bit, just do a couple of light passes until you hear that's it's not cutting wood anymore. Don't go too slow, you'll burn the wood, and then finish with a sander.
As for finishing, machine sand to about 180 or 220, and then hand sand with something higher, but don't bother going above the early 300's before poly. Only because birch is a bit soft, and can suck in the stain a bit unevenly.
Wipe it all with a damp rag, wait a bit, then polish out the raised grain with the same grit you used to hand sand.
For the edge grain (not the face or side grain) you may want to get some shellac (Zinsser clear) and thin slightly with pure alcohol. The shellac will seal the wood so it won't absorb as much, which is maybe what you want for the edge grain.
I would suggest buying the shortest length of 2x12 you can, use it for your router base, and use the leftover for practice for your routing.
Do at least one full test run of the routing process, preferably two, and use them as test pieces for your stain, since the softwood will be even harder to stain evenly than the butcher block.
If the edge grain is too dark compared to the face and side, add more shellac. If it's too light, either thin the shellac more, or sand some of it back off before staining. Practice on several test pieces, as much as it takes, figure out the right formula, practice on a scrap or two of the actual butcher block, adjust your formula, then stick to that formula with your finish pieces.
