I think I missed a detail in my understanding of missing residues. I understand that in some cases residues don't end up as part of the PDB model because of crystallization issues. However, aside from that case, will numbers be missing in the sequence of residues even though the residues are contiguous for the same reason that insert records may be used in the opposite scenario? In those cases, during processing they should be treated as contiguous?

Is this part of the reason why software packages such as Bio3d offer features for detecting chain breaks? Is this sort of software applicable to distinguishing the above cases?
Yes, as you wrote, the residue sequence number in the PDB format doesn't need to be sequential.
It's well explained by F.C. Bernstein here:
https://lists.sdsc.edu/pipermail/pdb-l/2004-March/001513.html
And some atoms or residues may be missing from the model.

How to distinguish the two cases?
If the PDB file has the SEQRES record - it should contain the list of all residues in the chain -- also the ones missing from the model. So this is one way of checking for missing residues.
Alternatively, you may check distances between atoms.
(I don't know anything about Bio3d).