Is there data supporting the claim that these traits correlate with selecting "better" candidates?

What evidence is there that compulsory voting wouldn't just add noise to the selection process?

There are about two dozen countries that use compulsory voting. Looking at the ten countries that actually enforce it we find that it in fact doesn't just add noise to the selection process. We find that they in fact don't have a selection process particularly dominated by noise.

If we look at actual compulsory votes, and find that practically nobody votes for some candidates while others get a lot of votes despite the addition of the reluctant voters then that which was added can't have been just noise. In this example only 1.4% of all (primary) votes went to the "Family First" candidate. Even assuming zero of the voluntary voters voted for "Family First" somehow the additional "noise" still knew to favor the other three candidates and mostly avoid Family First. Additional votes made by constituents and which systematically favor one candidate far above another aren't called "noise", they are just called "votes".

What evidence is there that compulsory voting wouldn't just add noise to the selection process?

I happen to know Mortimer Q. Snodgrass, who lives at 128 Ordinary Ln. is well informed about politics and economics but chooses not to vote because he is personally better off staying at home to play catch with his son. Obliging him to vote introduces information into the process that is not noise.

By the above I mean to reject the framing whereby "it would just add noise" is set up as priveliged hypothesis that significant 'evidence' must be expended just to escape from.

If the new voters actually "just add noise" it would mean every single one of them had no preference or information about what they are voting on and all choose to vote using a noise source. Even if we weaken the hypothesis to "the amount of noisy votes added relative to signal votes would do more harm than good to the preference extraction process" it is still an overwhelmingly unlikely proposition.

The basic voting systems just aren't especially vulnerable to noise, they are vulnerable to corrupted or biased input (some of which they cause themselves). If even 1 out of 20 of the new votes represents information about population preferences while the other 19 are 'noise' the system still ends up better informed. Noise just isn't the potential downside here, the respective biases held by the people who would have voted vs those that wouldn't and how much you endorse the groups and biases in question. It is at least plausible for someone to declare that the preferences of the demographic that would vote voluntarily are more important for some intrinsic or instrumental reason than the preferences of the demographic that must be compelled.

Even if a poll of all eligible voters is more noisy or just worse than a poll of typical voters, compulsory voting changes the game from the point of view of the politicians. In the US, convincing people on your side to vote seems to be a lot more effective than convincing people to switch sides. Compulsory voting changes the relative value of the two strategies, perhaps making voter turnout irrelevant. I don't know if such a change would be good or bad, but it sounds big to me. I also don't know what campaigns in actually existing compulsory voting regimes look like.

What evidence is there that voluntary voting doesn't just add noise to the selection process?

That's a serious question: voluntary voting means that a higher percentage of the voters are in a blue-vs-green mindset (since they are more likely to vote than someone who has weak preferences), while compulsory voting gives a more accurate picture of the feelings of the entire population, even if that involves people who donkey-vote etc.

(That's not to say your point isn't valid, just that the sword cuts both ways.)