If the turbocharger wastegate was not operational because of a loose hose, you would experience full boost from the turbocharger with nothing to regulate it. This would likely damage the engine before the turbocharger. The turbocharger is designed to spin all the time with the wastegate opening to relieve the excess pressure instead of it being fed into the engine.
The turbocharger could have been damaged by excessive heat depending on how long it was driven like this. Many vehicles have boost protection built into the programming in case a drastic overboost is detected. As a matter of fact my 1989 Probe's turbo wastegate has a tear in it. It does not work at all, however when my ECM detects too much boost it shuts down engine operation to avoid failure of the engine.
What is the current mechanic saying is actually wrong with the turbocharger.
Where was the smoke coming from... the engine compartment, the tailpipe or someplace else?
It is a possibility that with the wastegate non operational the trubo could have been overheated from the constant boost. The heat could have destroyed an oil seal, that would have caused oil to enter the exhaust and cause excessive smoke. Althought this is possible, I personally would have expected to see engine damage occur first.
Is the turbocharger now seized up? Has the impeller been damaged? Is their shaft play? There are many different failures that can occur inside the turbo.