Harry Clarke, 58, was driving the council truck in Glasgow city centre on 22 December last year when the fatal crash occurred.
Witnesses reported that he appeared to lose consciousness at the wheel.
A fatal accident inquiry at Glasgow Sheriff Court has heard claims that Mr Clarke failed to disclose information about his medical history on several occasions.
The inquiry has been told that Mr Clarke has a history of health issues such as fainting and dizziness dating back to the 1970s, including fainting at the wheel of a stationary bus in April 2010.
The inquiry previously heard that Mr Clarke had his car and lorry licences restored in April this year but both were revoked again in June.
Witness cardiologist Professor Andrew Rankin saw Mr Clarke as a patient in February and had some records of the 2010 fainting episode.
He told the inquiry yesterday: “I told him in my opinion he should not drive HGVs.”
Solicitor-advocate Ronald Conway, representing the family of Stephenie Tait, who died in the crash, showed the inquiry a DVLA log relating to Mr Clarke’s contact with them.
Mr Conway said: “What that would appear to indicate is, notwithstanding the terms of your discussion, by 2 April he is already seeking reinstatement of his HGV licence.
“That would be in direct contradiction to your advice.”
Prof Rankin agreed he had advised Mr Clarke that in the circumstances he “shouldn’t be driving HGV vehicles”.
Mr Conway said: “On the face of it he seems anxious to have his HGV licence restored.”
The professor replied: “He might have misunderstood me in terms of my suggestion that he clarify with the DVLA the diagnosis.”
The witness was asked his opinion on the duty of patients to “self-report” any fitness to drive issue to the DVLA.
Mr Conway said: “Can you agree with me that the idea of self-reporting may be a weakness in the system?”
He replied: “I think it’s a recognised weakness in the system.”
The inquiry also heard from specialist Dr Ronald Neville, who produces GP expert reports.
He said the current DVLA system was “not fit for practice”, adding it was possible for a person applying for a licence “to not disclose very important things”.
He added: “Unless this goes through the GP in some shape or form there’s a very, very real risk of a medical condition not being disclosed.”
Asked about the cost implications of any such change, he added: “I wouldn’t want to put a cost on public safety.
“Whatever it takes we have to make sure that persons driving our buses and lorries are entitled to do so and safe to do so.”
Earlier, the inquiry heard that Mr Clarke told Prof Rankin at their February meeting that he had not lost consciousness during the 2010 incident.
During cross-examination, Dorothy Bain QC, representing the family of victim Jacqueline Morton, put it to the witness that Mr Clarke had given him a “completely different” account of the 2010 incident to that given to other doctors.
“He told me that he had not actually lost consciousness,” the professor told the court.
Prof Rankin agreed that he appeared to have been given a different account to that given to other doctors, that Mr Clarke had in fact “blacked out”.
Mr Clarke has been suspended from work by Glasgow City Council “on a precautionary basis’’ ahead of a full investigation.
The Crown Office said in February that no criminal charges would be brought against Mr Clarke in relation to the crash but relatives of Ms Morton have called for him to be prosecuted.
Ms Morton, 51, and Ms Tait, 29, both from Glasgow, and Gillian Ewing, 52, from Edinburgh, were killed as the lorry travelled out of control along Queen Street and towards George Square before crashing in to the side of the Millennium Hotel. Erin McQuade, 18, and her grandparents Jack Sweeney, 68, and Lorraine Sweeney, 69, from Dumbarton, West Dunbartonshire, also died from multiple injuries after being hit by the truck.