Actually, we've been in business for 6 years and have a fully functional shop with all the equipment except a drill press.
-Not sure what "fully functional" means, perhaps functional enough to work on golf clubs. What I'd suggest will depend upon the level of work you want to set for your customers. If you want to offer garage hobby shop level of service that's one level. Offering a professional level of service to expensive clubs dictates another level of service. Not knowing your clientele, I can't estimate what their expectations would be. Were it me or a semi-professional golfer, if I saw somebody attempting to work on an expensive club head with a drill press I'd brand them as hacks. That's what some of the pro shops use? That's because they're hacks with machines that don't know any better. There are a lot of people that think they're machinists just because they're not bleeding at the end of the day. I'm not the last word on machining but do have 35+ years of shop and design experience. I'm not advocating a spending spree, I'm trying to get a good understanding of your operation so I can give you proper advice. No point in going butterfly hunting with a mallet any more than using amateur methods for professional grade workmanship.
Looked at HD & Lowes and scouted CL & FB marketplace, but don't know what brands to avoid or investigate. We now reside in central Georgia.
-That's ok to view what's readily available but HD & Lowe's only carry light duty, low quality drill presses. I looked at several models (4) on their website and think none of them are what I'd use for reaming an expensive part. Since you're now in Georgia (you should have stated this earlier) there are likely greater options than OZ. CL and FB are good to keep checking but there are machinery dealers that carry used industrial grade machinery that's still in good condition, none of my machinery was purchased new. Sometimes you have to look for a few weeks and be willing to drive an hour or two to get it.
Not sure which style the golf club reamers fall into,
-The type of reamer is all about what type of hole needed to fit the replacement shaft to. Is the end of the shaft (that goes into the club head) a tapered fit or is it a constant diameter? What type of hole do you need for fitment? Are you altering the replacement shaft to fit the club head or altering the club head to fit the shaft?
but the majority of operation will be boring out hosels from .355 to .370.
-Not to be overly critical here but boring is a different type of operation than merely drilling/reaming. Drilling means lining up the prospective hole location and having a go at it. The drill can wander, deflect, and end up off location from where you intended it to be. This is especially true if the hole has a broken shaft still in the hole, the hole isn't exactly aligned (critical process) with the drill axis, or there's trouble with cutting the club head material. Under certain conditions you can drill a hole out of center or end up breaking out the side of the club. Both will be noticeable to the customer. A reamer will tend to follow the hole even if it's off location. That's just the nature of drilling regardless of what anybody says. The fixture for holding the club head also makes a difference but that's a different discussion. The unguided fixture in the photo is questionable and may not produce the results you want. Under some circumstances I can see it failing you and ruining the club head.
Boring uses a different method, usually a single cutting edge tool is used instead of a drill bit. It will make an exact hole size exactly where you decide it should be. It doesn't deflect or wander like a drill bit, gives more accurate results, but is a more involved method. Both methods are appropriate, you have to know when to use which one for good results.
I'll say it again, light duty hobby shop equipment can produce decent results if you know how to compensate for the inadequacies of the equipment. You also stand a chance or ruining expensive club heads now and then by using hobby grade equipment. Just because some of the Pro shops use hobby grade equipment doesn't mean they're producing professional results. None of the pro shops use a mill because they probably don't know how to use one. Your expectations for the type of business you want to run are entirely up to you. Expensive equipment doesn't guarantee professional results either. Spend time to understand the process before you spend your money. Want more details? Post more details. Have questions? Ask questions.