I've rewritten some PQ implementations that had RCEs and memory disclosure vulnerabilities in them. No shade, but those implementations were from scientists who don't typically build production systems. As an industry, we're past this phase. Side-channels more commonly reveal plaintext than key material, but that shouldn't be fatal in the case of hybrid key agreement.

Based on what we've seen so far in industry research, I'd guess that enabling Denial of Service is the most common kind of issue.