Attached is a PPT in PDF form that covers a good amount of ground on X.509, PKI, and TLS/SSL.
All Browsers will validate a chain, but when you go to find the chain, the browsers will pick the first certificate based on the Distinguished Name. Many CA cert vendors are re-releasing 'same-named' CA certs, so the chain can be a 'false chain'. Why is this? It is cryptographically cheaper to parse a public key and certificate than it is to validate the signature, and it is not always possible to trace serial numbers, so Browser vendors look to the DN/CN and pick the first one they find...Bob is Bob, even if the DNA is different? No.
Sites are not under any obligation to send the full chain. I have many examples of partial chains, usually missing the self-signed ROOT.
Some sites are 'rooted' (pun intended) with a very old CA - X.509v1-based - and modern infrastructure may reject them for valid security reasons.