Do all banks rely on intermediary banks for international wire transfers?

No.

If two banks have direct access to each other and have a settlement arrangement between their ledgers, then needless to say an intermediary is not required.

If both the banks are the same bank, think Deutsche NYC and Deutsche London, then an intermediary might now be required as it would be a book to book settlement.

Intermediaries are used when the originating bank and the settlement bank do not have a direct arrangement with each other for that specific foreign currency, or if mandated by law to settle through an assigned 3rd party.