I don't know who has been filling your head with such poppycock, but the statement in bold is pure nonsense when it comes to places like Catalina Island, various small islands of British Columbia, and the islands off New England. Whoever put forth this theory must be living in the 18th century, because today we have such things as helicopters and planes to air-evac any ill or injured person to the mainland for proper medical help. This would be the case for a local resident as well.
As for places like the Caribbean, where air-evac isn't as feasible, they certainly wouldn't want non-citizens filling up their hospitals, but so far the infection rate down here has been very low, so they probably have room for foreigners, too.
The reality is that borders are closed, and islands don't want visitors, not because they can't deal with the infected, but because they don't want outsiders bringing the virus into their community. And rightly so, IMO. Some may think that sailing is the ultimate isolation, but getting to and from the boat requires passing through and into society for food, parts, fuel and if needed, help, that wouldn't be needed if one were isolating at home.
Every time someone leaves their home they expose themselves to the virus, and if they contract it, even if they don't become visibly ill, they could pass it on. IMO, that is the equivalent of manslaughter. It is why those protesting to reopen the country too soon disgust me. Most are young enough that they may not become terribly ill, but they have shown that they don't even have the basic intelligence to wear a mask (since their leader won't, why should they?) or social distance when out in a large crowd, so once infected, they will be a danger to anyone they come in contact with