I’m from the UK so I’ve never done the tour operator thing but thought I would give you some info to get you started.
There are essentially 3 golfing hubs in Scotland that you’ll want to look at.
The first is, obviously, The Auld Toon, St Andrews. Other than the Old course, there are several others, including the New, Jubilee, Eden, Castle, Kittocks and, just down the road, Kingsbarns. Carnousite is just up the road and easily drivable.
The second is Lothian, which run along the coast east of Edinburgh. This has perhaps the biggest concentration of world class links courses, from Muirfield, North Berwick and Gullane, as well as newer course like Doak’s Renaissance.
The third is Ayrshire, which runs south west away from Glasgow on the west coast and includes Turnberry, Troon, Dundonald, Prestwick and tracks like Western Gailes.
Any one of these regions is worth a full 10 day trip on it’s own, but as you’re looking to do the whole lot you can drive from one to the other in 2 - 4 hours, depending on traffic. It would be possible to base yourselves at just one hotel for each of these 3 locations, so you’re only base jumping twice. You can fly into either Edinburgh or Glasgow and get a rental car, with either of the 3 locations only a couple of hours away. Do the car rental in the US before you leave though, as its much cheaper.
As for hotels, you’d need to do some research. I don’t think any of these regions are spectacularly well served by the big chains (IHG, Hilton, Marriott etc), though you could choose to stay in Edinburgh for the Lothian segment and you’d have every hotel there. It would just mean a 1-2 hour drive each day to the course. There is a Fairmont just outside St Andrews (the Kittocks course belongs to them) but I don’t know if there are any other big chain hotels in the town. Booking hotels is pretty straightforwards though.
The biggest issue is the cost of playing and whether a company can get you better deals. Most courses have websites so just look it up and cost it out. It could be worth an email to see if they could do you a deal but don’t expect it in high season. Compared to the US most UK courses, even the Open venues, are pretty reasonable. You’ll need to work around Muirfield as they only allow visitors on certain days but other than that they are mostly accessible. Also, book the Old Course, but if you want to spend another day or two there you can get a 3 course pass for about $200 and play on 3 courses which are chosen at random by ballot each day.
Outside of those three areas there are some spectacular courses that are more isolated. Royal Dornoch is one such example, but it’s near Inverness, which is a 5 -6 hour drive from St Andrews. Also up there is Castle Stuart but that’s about it (funnily enough Mike Keiser is trying to get permission to build another mega complex right by Dornoch, which, if it happens, would turn that whole area into a major golf destination). There is Trump International near Aberdeen, if you don’t mind giving your money to the orange faced fella. Machrihanish and Machrihanish Dunes (designed by David McLay Kidd) are worth a trip but it’s a seriously out of the way spot hours from anywhere.
Also, just so you know, there is a ferry terminal at Cairnryan which is about 30 mins from Turnberry. From there a ferry to Belfast is only a couple of hours and both Royal County Down and Royal Portrush (which will hold the Open next year) are about an hour away. Might be worth considering as a little bonus for a day or two.