By your dichotomy, Physics seems closer to Psychology than math. Why aren't there replication crises in Physics? My answer is that there always has been-- it's just that they get resolved quick. The problem with Psych is that it is SLOW
The entire point of the replication crisis is that we should expect statistical anomalies to occur, and the problem is that anomalous results are wildly more likely to be published. Which is why you probably hear about evidence of life on Venus but not about calibration issues.
It’s just generally easier and more problematic to do experiments testing framework boundaries in sciences with more variables like medicine, or anything to do with behavior.
Testing a new drug, for instance, is very difficult because there are about a billion chemical processes going on in the human body (or whatever body is being tested), and trying to distill the effects of adding one more, and making sure that the effects are correctly attributable to the right causes is something we can almost certainly never know 100%.
This kind of thing certainly happens in physics, and we can see results like neutrinos appearing to travel faster than light until they are corrected, but it's usually much easier to isolate the processes of interest in a physics experiment than in a biological system, or at least easier to eliminate most extraneous effects.
Psychology will always have ethic, sample size and cultural/time/space constraints. Your "psychology experiment" done with Mechanical Turk is more an exercise on sampling bias than anything else
And they keep running into "replication failures" because they fail to account for that and try to chase the illusion that it is possible to have a perfect controlled environment in psych. They can't, and they're only fooling themselves they can.
The problem with Psych is that you can't really experiment on humans in a meaningful way. Animal studies for example don't suffer the same replication problems as psych.
While absolutely valid this isn't the only problem: One other complication is that conclusions from experimental studies often don't apply to the real world, i.e. people behave differently in experimental settings. Thus in any case you can either have good control over confounding varables but very limited scope (experimental studies) or bad control over the setting but the ability to observe real live behavior (field studies).
But it gets worse: Two people with identical behaviour can have quite different motivations and thus identical observable behaviour can result in quite different consequences. I.E. 5 hours a day playing games can be both healthy behavour or an addiction/a means of suppressing thoughts or memories, depending on the person. So you'll always have that massive confounder called the mind, in every setting. Which means you have to inspect the mind itself, which only works indirectly, i.e. by interacting with the person, asking questions, trying to find out the motivation behind a particular action. And the person often doesn't really know, because a particular decision probably wasn't chosen consciously.
[0] Guess when I was experimented on? Of course it was during university and of course I'm a white english speaker that lives in North America.