The answer seems a bit obvious. Am I missing something?
What does Amy Coney Barrett have to do with abortion, healthcare, etc?
As a Supreme Court Justice she will have to make decisions on a broad range of issues, including abortion, healthcare, gun rights, etc.
Why would Democrats - or anyone, for that matter - grill Amy Coney Barrett on abortion rights, healthcare laws, guns, etc?
First of all, because they can. Article II doesn't go into detail on what "the Advice and Consent of the Senate" mean. They can ask nominees whatever they want and they can vote based on character and political beliefs. To give an example, the significant part of Brett Kavanaugh and Clarence Thomas hearings was dedicated to the allegations of sexual misconduct on behalf of the nominee.
Another reason is that those are the issues people care about and take personally. For senators, public hearings are a way to inform their constituents about the court, and the impact on their lives. And some of their constituents are interested to know whether future Justice Barrett will vote to strike down ACA or to ban abortions. And if her personal beliefs will influence her decisions as a Supreme Court Justice it makes sense to ask her about them as well.
For the same reason, Republican senators grilled Elena Kagan on same-sex marriage, abortions, and enemy combatants.
Instead they ought to be grilling her on her legal experience, her criminal record (if any), her ability to work with the current justices, her personal health
There's not a lot to discuss. Amy Coney Barrett was already confirmed to serve as a circuit judge in 2017. We can assume that if there were disqualifying issues with her legal experience, criminal record, and mental/physical health they would have been uncovered during those hearings.
It seems to me like if the government needs to know Amy Coney Barrett's political beliefs before appointing her as a supreme court justice, then it might as well be a directly elected position.
This isn't really necessary. The Constitution already requires her to receive the approval of directly elected politicians. And the public definitely needs to know Amy Coney Barrett's political beliefs, especially if they influence her decisions as a judge.
Like, if one does is against abortion and can muster the required majority in Congress to make it illegal, then it doesn't matter what Amy Coney Barrett's political beliefs are because she will have to impose the law. ... Congress could just pass laws that leave no room for interpretation.
It seems you're trying to argue that SCOTUS doesn't have any real impact as Congress can just rewrite laws over and over again. Evidently, this is not true, as SCOTUS substantially changed the interpretation of the law many times (see Obergefell v. Hodges).
Also, to "muster the required majority in Congress" is not an easy task. You need the President and the rock-solid majority in both houses to pass the law and plow it through a hostile SCOTUS. Lately, neither party was able to keep this kind of domination for more than a couple of years.
we would expect Democrats to be grilling Amy Coney Barrett on whether she's an Originalist or Living Constitutionalist, not her political beliefs directly
No one would expect that. It makes sense to ask her about the potential decision she'll make if confirmed. And "I'm voting against Judge X because she will ban abortions" is much easier to understand than "I'm voting against her because I have a theoretical disagreement with her on the subject of statutory interpretation".