So starting with your title question :
Why aren't the Justices of SCOTUS democratically elected ?
Well they are democratically elected, but it's not a full plebiscite.
What happens is, as you know, candidates are proposed and they are vetted and either accepted or rejected by the people's democratically elected representatives.
Why it was designed that way is, I think, that to select someone with suitable qualifications it was thought necessary to have the candidate reviewed by well educated and knowledgeable people. Remember that at the time the US constitution was created the average person would have had little education, practically no knowledge of law or what defines an expert lawyer from a run-of-the-mill lawyer. The logic goes that, in that day, the people most likely to perform that function were your elected representatives who could be expected to have a reasonable education, a solid interest in law (even if it was in many cases just to break it themselves :-) ) and who had a mandate from the ordinary voter to make decisions for them.
Changing that requires a constitutional change, so unless that happens the ordinary US voter is stuck with the system they have courtesy of the supposedly all-knowing founding fathers.
In following the aftermath of Justice Ruth Bader Ginsburg's death, I notice the process for appointing a new justice appears to be very politicized. I don't understand why.
It's a democracy - everything is politicized. It would likely be even worse if a full plebiscite were required as it would be like a super-sized version of a presidential campaign (for a for-life appointment to a position of considerable power !).
If it was just a straight decision of one man without oversight (as I'm sure many a POTUS would have preferred) then it would not be politicized as much as people would effectively have no say at all.
So while it may look politicized now, that's just because it's got some democracy in the mix.
I would have thought justices decide on purely legal questions: X is either legal or illegal. Y is either constitutional or unconstitutional. Those are different questions compared to "X should be legal or illegal", which necessarily involves personal opinion.
This is probably a question more suited to the Law SE site.
My very limited understanding is that SCOTUS does not have as well defined a role or powers as people may think it does. Much of what it does today seems to be defined and limited by it's own ruling - by precedent. States respect those decisions but it's not at all clear to me that SCOTUS can enforce arbitrary decisions.
Should X by legal or illegal, e.g. like the abortion cases, is a matter they seem to have had as a defined right as long as they are acting as a final arbitration of interpreting the application of existing laws and what priority those laws have over each other. They are ruling on whether laws are being applied in accordance with more important rights and laws as defined by existing laws and the constitution.
They do not, as such, make a new law. They have to be asked if they are willing to decide if laws made by the US, State or local government (or law of Tort) are being reasonably and fairly applied in the larger context of the full rights and laws applicable. That is not making new law, but if you are suddenly told by SCOTUS that you can no longer do X because e.g. lots of pages of detailed legal logic, it will seem like a new law. But it is just existing law being put in it's proper context.
However note that as the legal landscape they operate in (the big picture of all laws and amendments) changes over time, and as social norms change over time and finally that the individual members change over time, SCOTUS will inevitably seem to "change their minds" as time goes by. They may hear a case in one decade and decide that, hey, X seems right, and then a decade later, hear the same thing again but framed by new laws and social norms and decide Y is right and X isn't right now. What was perfectly reasonable and fair as a method of execution one decade may be considered unfair and downright cruel by the court in another as law, social norms and maybe even scientific understanding (e.g. new evidence) develop. So it's not as black and white as may seem.
the government doesn't have a choice, they have to make it legal for women to have abortions without excessive government restrictions
Governments do have a choice. They can try and pass new laws or amendments to the constitution, get them passed and wait for the inevitable appeal to SCOTUS to rule they got it wrong, hoping that this time SCOTUS feel they did it right and did not trample on more significant laws and rights.