I have to concede you've introduced a line of thought I hadn't gone down. But whilst I pull on that thread I come to a barrier at the point of an interpretation of what a right is.
My personal opinion is probably much colder and harsher than most, I don't believe that "rights" exist. I don't believe anyone truly has a right to anything, I only feel there is accepted rules/regulations and there is enforcement. Why do I not kill someone? It's not because I believe they have a right to life, it's due to fear of consequence, I don't want to live with those consequences so I don't perform those actions. To me rights are a false sense of security and or entitlement you hear it all the time "you can't do that it's against my rights" ok, sure, watch me, if that person doesn't have the capacity to prevent me from infringing on their rights then are they rights? Or are they certain expectations in behaviour that can at any moment change. Using the right to life as an example, if it's a basic human right, then how is it ok to shoot and kill someone? In warzones? Or even defending your own property. If your right infringes upon their right, then theyre not basic rights they're rules that get followed most of the time and the rule that gets followed there and then is the one that can be enforced by whoever has the most power over the other.
In the example of the UAE the govt has not removed someone's right, by the definition you're describing they're just preventing them by an abuse of power from being able to exercise this so called right.
In your island situation, which I found very interesting as a thought experiment, without the govt there is no requirement for enforcement of rules. There is just an expectation that a small collective of people will behave in a manner in which they wish to be treated. I'd suggest they weren't not killing each other because of any perceived rights but instead out of either a fear of consequence of getting killed or exiled by the rest of the community. In that situation the collective needs each other to survive. They're not agreeing rights they're agreeing behaviours that gives them all the best chance at survival.
Just to attempt to articulate my point on rights not being something I believe to be real, I'm going to use the term entitled instead of rights. let's roll back to guns again and even yes use healthcare like you mentioned, instead of claiming to have a right to healthcare, you are entitled to access, but there is a caveat... And that caveat is that you meet acceptance criteria. In the case of healthcare one of the criteria is that you can pay for the care. That's not a right if there's a barrier to entry.
So I guess what I'm getting at is no, you aren't entitled to own a gun or have a right to own one. There are barriers to access already in place, what I'm saying is those barriers should be tightened and bring in a level of necessity. Nobody that is not engaged directly in military conflict, needs military tools. It's a risk to reward weigh up, the risk of the wrong people using those weapons outweighs the reward of anyone having access to it. Just to elaborate a touch, the risk to the lives of multitudes of children and adults because that weapon came into the wrong hands, far outweighs the reward of one person being able to take down one aggressor to defend themselves, their family and their property. It's a simple numbers game. Let's say average family is 4 people, the wrong person decides to start shooting at you and they get you and yours. 4 deaths. Or you get to them first 1 death. Your access to that weapon had a potential for up to 5 deaths (including yourself or the aggressor, situation dependant). Now looking at that same weapon in the hands of an unstable person in a school, 40+ easily. The maths doesn't check out. The needs of the many outweigh the needs of the few and the many need to be alive over the fews want to own a gun.