Consider rewriting Schema-compatibility verification (`SchemaCompatibility`)

As per title: current verifier seems to both be overly sensitive (possibly false alerts) AND also does not indicate what exactly went wrong (just that it did). It should be possible to improve on this, rewrite the logic following the baseline checks but not necessarily implementation.
As an added bonus, it should be possible (if we choose to) run verifier for multiple issues, not just the first one.