How to ensure that a cryptographic implementation is reasonably sound/robust? I know from the bottom of my heart how bad the idea of custom-crypto is. I absolutely concur with the recommendation that it should only be done by professionals who specialize in implementing crypto safely.
However, exactly because it's a Bad Idea ™ it's really hard to find good information on this topic. Say one of the aforementioned professionals is looking at a crypto implementation and they want to judge how safe it is. Some things that come to mind:

Constant-time implementation (to protect against timing/side-channel attacks)
Randomness

Source of randomness (true vs. pseudo RNGs)
Bits of entropy


Zeroing intermediate buffers
Appropriate test coverage with test vectors (e.g., NIST, Wycheproof)

What are other areas like this that need to be probed? Is there any documentation/checklist/book that can shed further light on the topic -- something like 1 but more comprehensive?
One widely accepted standard for information protection is the ISO Common Criteria framework. This set of slides gives some sense of what a Common Criteria evaluation will ask in its analysis.
The US FIPS-140-3 standard also carries a lot of weight, it specifies certain NIST special publications that give criteria beyond ISO.