Engaged tone comes from the exchange local to the caller. If there is a fault it may be played anyway. The problem is to isolate the fault and see if it is on your premises or off them.
It seems to me that removing all the phones is a good first step. I'd suggest opening up the main socket (the one the incoming line comes in on) and seeing if you can disconnect the extensions. That's what I'd try anyway.
Then I'd try connecting a trusted phone directly and seeing if the problem still exists. If it does then the likelihood is that the phone service supplier hasn't checked properly and you and insist they do so, reasonably confident that they aren't going to chase the problem back to your property and start charging.