Ok firstly you are completely wrong on not only how SWIFT functions but also basic understanding of the USD wire instruction you posted in the link. So let me explain below.
I will begin by saying that anyone who reads the wire instruction and understands wire messaging can determine that the bank is connected to swift at first glance. You started by quoting the wire instruction for an ABA not SWIFT transfer.
Did you notice that this is the ABA wire instruction for an entity of Caye Bank in America and not the bank in Belize? The address and name of the account is big clue as they are both in America and not in Belize. This is an account held at First Savanna Savings Bank in America. That particular instruction is for US ABA transfers and has nothing to do with international SWIFT wires. If you looked at page 2 on same document you can see the actual SWIFT USD wire instructions for international SWIFT wires where you would enter Caye Banks SWIFT and crown agents banks Correspondence account details on the wire instruction.
How SWIFT works is that their is a huge data file containing all the BIC's. There is also a routing file that banks load into their static data on their IT systems that contains all the routings for BIC's on a per currency basis. So for example if I entered the BIC CAYEBZBZXXX and the currency i.e USD you would immediately be given the appropriate correspondence bank that CAYEBZBZXXX will route that currencies payment to i.e CRASGB2L (which would in turn use a correspondence bank of its own in US). In reality if static data is kept up to date on sending banks systems I would only need to enter CAYEBZBZXXX and my account name and number and the payment will be routed correctly to Caye Banks USD account at Crown Agents correspondence bank. The problem is that not all banks update their static SWIFT data regularly enough. Some do it in real time and others maybe once a month. This means the receiving bank like Caye will always advise you to enter the correspondence bank manually into Field 56 of the MT103 message to make sure this issue is avoided. You do not want to rely on default routing in the sending banks static data being wrong. So if they were not connected to swift then why are they publishing a USD SWIFT wire instruction with CAYEBZBZXXX as the beneficiary SWIFT address?
Ok they are using for correspondence banking for other currencies from Crown Agents bank system. They would typically just be loading outgoing wires from a spreadsheet into the IT system of Crown Agents bank to affect the outbound wires and likewise to receive the inbound wires. There is typically no need for them to have any SWIFT overhead for sending and receiving funds outside USD. I can imagine the bulk of their business is in USD hence why they offer two sets of instructions for USD for redundancy. Again I am just going by the USD wire instructions they publish on their site and assume them to be correct. If they are not up to date and the instructions on page 2 are no longer valid. then I am wrong fair enough
Hope this was helpful to someone.
The wire instructions they have given are concise. Unlike other wire instructions that hide all the detail involved in an international wire instruction. 
