Looking for a private secluded Carribean cottage with beach (and pool?), small boat. Car or golf cart for travel to explore and shop. Does it exist?

Kinda, you can always rent a villa on most islands (car rental vs golf cart though)

Out Islands in Vahamas have that

But—

Keep in mind while property is private, all beaches are public and so there’s always someone walking by

What kind of price budget? It comes with a cost

How “private” & “secluded” do you really have to be and what’s your definition of “cottage”.

Beach front property in the Caribbean is very expensive so most people who can afford the price tag aren’t going to build a small one or two bedroom “cottage”. Instead they will build multi-million dollar multi-bedroom high end villas.

Also, seclusion comes at a price, the more secluded a place is the more expensive it is to build which again translates into affordability by more affluent people who don’t build tiny “cottages”.

Now, many mid-size (say 3 to 5 BR’s) villas can be
rented for a lower price if you guarantee that there will only be 2 people in your party so that is a possibility but on most islands the rental villas will be in close proximity to other properties.

Most beachfront rental properties won’t have boats since there are usually local laws limiting docks on beaches plus there are liabilities concerns that deter owners from offering renters use of boats. You can, however, rent boats on many islands but you probably won’t be able to keep them “on the beach” by your rental villa. Of course, if you rent a “cottage” that is part of a bigger resort you may be able to dock your rental boat on resort property but the trade off will be loss of privacy/seclusion.

It would help if you gave us some idea of your rental budget, when you’ll be traveling, where you are traveling from and how long you are willing to travel to reach your final destination.

As blamona says your best options will be on one of the Bahamas Out-Islands since they are sparsely populated and tou’ll Have a better chance of finding the seclusion and privacy you are seeking.

Travel from S. Florida will be next July. Since this will be our 50th anniversary, I will pay what it takes to get what I am looking for.
Regarding privacy, some beach traffic is fine; but looking for total privacy by the house and pool. And a good bar/restaraunts in town would be a real plus.

Check villa rentals on Cat Island and Long Island, Bahamas. Both sparsely populated quite islands - with a little research you should be able to find what you are seeking.

I would suggest you look at Eleuthera Gov Harbour area. You should be able to find a cottage on a 'private' beach but not sure if any come with a boat.

Cat is private but dining and shopping is very limited.

Long island is lovely. Check out Whymss villas on Long Island, Bahamas - directly beachfront lovely homes and come with a boat. You should be able to find plenty of reviews online.

https://www.homeaway.com/vacation-rental/p173210