Mortgage insurance company requesting blood work, won't tell me why

This happened a few months ago when I renewed my mortgage and it still bugs me, and I'm hoping someone can tell me what the point of it is. Because the insurance company wouldn't ('couldn't') tell me.

I renewed my mortgage, technically ported it to a new lender because the original lender too expensive, and in the process they asked if I wanted to continue with mortgage insurance. I said yes, but just basic AD&D in case something happens to me then it will pay off a good chunk of the remaining balance. They pushed me to get the health coverage, which I did had previously, and I said sure, it wasn't that much more and it was good to have the peace of mind. When it came time to sign the paperwork they said that they just needed bloodwork, and I asked what for and let them know that it might be a while before I can get in to my doctor for a referral for bloodwork. That's where things went wrong.

They said they have their own people that come collect blood and do the testing, no bloodwork requisition needed. Here is basically how the conversation went; M = Me, I = Insurance company.

M: Who takes the blood, someone from the hospital?
I: No, we have nurses on contract to do it. They come to your home and take your blood to do testing, and then when we have results back we confirm everything.
M: Okay, what is the testing for? What are you checking?
I: We test for different things.
M: Like what?
I: It's a different situation for everyone, and we really can't say what it is because it's so different.
M: I don't agree with that, if you think there is something wrong then I should know and have my doctor involved. What if you find out I have an illness and I don't know about it?
I: That is very unlikely based on the kind of tests we do.
M: Oh okay good, so what kind of tests do you do?
I: Like I said, it's different for everyone.
M: Okay, who has access to the results and what are they used for?
I: We keep them on file.
M: For what?

Anyway, I ended up in a circular discussion with the insurance company about why they want bloodwork, what they thought was wrong with me ('nothing, we just do this' apparently), who would have access to the results, what they use the bloodwork results/specimens for, how long they are kept, etc. I spent almost 2 weeks going back and forth with them before one of the insurance agents finally told me that because I already had mortgage insurance through them and the original agent didn't link it to a specific mortgage account, I was covered at the original amount wherever I went. So basically, if I took out a bit of extra money when I renewed and I needed the insurance to pay, they would only pay the amount they expected to pay when I first got my insurance. But otherwise, I can take my insurance coverage as is (full coverage) and they can't request any additional testing.

So, this brings me to the question: What does the insurance company do with bloodwork results?

I'm not being paranoid here; I work in healthcare and I know how test results can be misinterpreted, misused, and how privacy violations can occur. I also had a suspicion that in the event I tried to make a claim for a health condition several years later, they could say "But based on your bloodwork you had markers for X illness, so it's not covered by the policy." but again, no one would confirm what the results were for, how they were stored, who had access, etc. And they wouldn't take a copy of the results or a statement from my doctor.