Yes, hooves & hair are the same material & it has been scientifically tested & found to increase rate of growth & strength. But to bsms, you wouldn't see any difference in only 4-6 weeks. It takes around 9 months to start seeing any change. Hooves grow at different rates for a number of reasons. Diet - so seasons - amount of exercise, nutrition, hoof health, etc all have an effect on growth.
OP, if you are already feeding a well balanced diet and the biotin is fed in excess of that, I wouldn't worry about giving it if there's no special need to grow hooves faster. If you're feeding it as the only supplement, I'd change to one that gives the horse all that he needs, rather than one single ingredient. I wouldn't stop giving it purely because you hope to slow hoof growth for financial reasons tho. If your horse needs trimming 4-weekly, that is not at all absurd and while 6 weeks seems to be the norm, I think most horses benefit from more frequent trimming, as it's always best to *maintain* the shape & function rather than wait for it to overgrow & deform before 'correcting'. Re boot fit, hopefully your trimmer will be happy to give you a couple of lessons in maintaining hooves between trims, so that you can 'brush up' with a rasp yourself.
I talked to my trimmer when she was out last and she doesn't think that stopping the biotin would be detrimental to her feet. She asked about her diet and feels that her diet is complete enough that she shouldn't need it, worst comes to worst I will put her back on it, it's only $13/month from smartpak but she is also on an ulcer supplement,a moody mare supplement and a pre/pro biotic, then alfalfa pellets and TC senior so with all that it starts to get expensive so if she doesn't need the biotin, I'd rather not pay for it. If it slows her hoof growth a little bit that would just be an added bonus
When she was at home she would go 8 weeks between trims and never seemed long, but since I started boarding her her feet seem to grow a lot faster.
ETA: My other farrier(does my other 2 horses at home) taught me how to rasp my horses feet, but without a stand I find it very VERY challenging, I've done it when I was having issues getting her boots on just enough to make the boot fit, but I'm still weary of doing much as I don't want to mess up her feet.