If the certificate chain cannot be built to a trusted root then none of
the keys can be trusted either. An attacker could build a totally bogus
chain using their own keys and valid signatures... but it would not be
valid because it would not chain to a trusted root.

Signature verification can be an expensive operation, for certain
algorithms and certain key sizes. OpenSSL does not check signatures
until the chain is complete, and then from root to leaf, to avoid a
possible DOS attack.
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