The family of a prominent politician who took his own life after being sacked amid allegations of inappropriate sexual behaviour have called for an apology from the former Welsh first minister after a coroner ruled no official support was put in place for him.

At the conclusion of Carl Sargeant’s inquest, the coroner John Gittins said he would write to the Welsh government warning that there could be more deaths unless better support was provided for high-profile figures when they were sacked.

But outside court Sargeant’s family focused their attention on the role of the former Welsh first minister Carwyn Jones, accusing him of letting his friend and colleague down and failing to show remorse.

Sargeant’s son, Jack, who has succeeded his father as assembly member for Alyn and Deeside in north Wales, said: “We are deeply offended by the lack of any remorse or regret from the former first minister. We very much hope that the former first minister will now come forward with a genuine apology.”

Jack Sargeant added: “I do feel that dad was let down by Carwyn Jones. Carwyn Jones said on the stand he was a friend. I can’t see myself letting down a friend the way he let down his.”

The family criticised discrepancies in the evidence Jones gave during the nine-day inquest and expressed frustration that Jones’s legal bills have been paid by the public purse while they have been forced to eat into savings to fight their cause.

Neil Hudgell, solicitor for the family, said: “We have seen a puppet master that has used the public purse to pursue his own ends without any regard for wider mental health considerations.”

Calling for changes in the way people are supported after accusations have been levelled, Sargeant’s widow, Bernie, said: “We never want this to happen to another person involved in politics or any other walk of life. There has got to be some safeguarding whatever your profession.”

Carl Sargeant, 49, was found hanged at his home in north Wales on 7 November 2017, four days after he was sacked from his job as cabinet secretary for communities and children. The inquest was told he had suffered from mental health problems and was vulnerable.

The inquest in Ruthin, north Wales, heard that Sargeant felt “shellshocked” and “destroyed” after his sacking. His family claims there was no support for him and he was left in limbo, not able to defend himself because he did not have details of the claims against him pending a full investigation. Bernie Sargeant said he insisted he was innocent and that she believed him.

As he began delivering his conclusions, Gittins told the court: “The twists and turns of the journey at times along the murkiest paths into the world of politics has been a challenge.

“Anyone hoping for a glowing vindication of Carl Sargeant or a damning vilification of Carwyn Jones, or indeed vice versa, will be sorely disappointed.”

The coroner said the truth of the allegations against Sargeant had not formed any part of his investigation. But he said: “The absence … of those allegations at this inquest should in no way be seen as a diminution of the seriousness of these matters.” Gittins accepted it was right for Jones not to give details of the allegations made against Sargeant or the identities of his accusers.

During the inquest Jones was accused by the Sargeant family of lying under oath over the care provided to Sargeant. Gittins said Jones’s evidence was contradictory, but accepted Jones had “erred” and properly corrected his error.

However, the coroner said: “No official arrangements were put into place by way of seeking to provide support to Carl Sargeant despite the probability that the first minister knew of Mr Sargeant’s vulnerability in relation to his mental health, and that it is likely that the removal and particularly the reason for the removal would generate significant media interest.” The coroner said he did not feel that Sargeant had received “pastoral” care.

Gittins said he had heard from the Labour party that changes had been made to its safeguarding procedures and the new Welsh first minister, Mark Drakeford, has said he would created an aide-memoire for himself to use when sacking ministers in the future. But the coroner expressed concern that Drakeford said he could not compel future first ministers to follow his lead.

Gittins also recalled an exchange he had with Jones during the inquest when they discussed the level of support that might have been given. Jones had asked: “Where does it end?” the coroner replied: “It shouldn’t end because we’re talking about people’s lives. It cannot end. It’s not a matter of law or duty of care, it’s a matter of humanity, that’s the responsibility of us all.”

On the court steps, Jack Sargeant said: “We recognise the ‘murkiness’ the coroner referred to in his summing up. It’s been a thoroughly distressing and dehumanising process that has added to our heartbreak.”

The family were angry when Jones’s legal team tried to argue that evidence about allegations against Carl Sargeant should be aired.

Jack Sargeant said: “At times it seems to have been forgotten that this was an inquest into the death of a dearly beloved husband, father, son and brother. Instead it has felt more like a criminal trial. All too often politics have been at play with the sole aim of blackening a dead man’s name to protect another. Where has been the humanity in that?

“We sincerely hope that no political family will go through what we have been through these past 19 months. It’s too late for Dad but may save someone else. We hope that political parties and governments across the union take note and make it policy for safeguarding measures to be in place for all public servants.”

Jones issued a statement in which he said: “This has been a difficult time for everyone, the family most of all and I offer them my deepest condolences for a loss that is inevitably still incredibly painful.

“The nature of these proceedings has meant that there appeared to be two sides in this matter, and whilst it is right that arguments are tested, the process has driven an unnatural wedge between people who remain united at the very least in their ongoing shock, trauma and grief. Nobody wanted this, and nobody could have foreseen it. Suicide is a shattering experience, and I hope some healing can now begin.”

A spokesperson for Drakeford said: “I extend my deepest condolences to the Sargeant family. I know they continue to grieve for Carl. The coroner has made a regulation 28 report [to prevent future deaths] to the Welsh government in relation to the prevention of future deaths and we will consider this carefully and respond in full.”

• In the UK and Ireland, Samaritans can be contacted on 116 123 or email jo@samaritans.org or jo@samaritans.ie. In the US, the National Suicide Prevention Lifeline is 1-800-273-8255. In Australia, the crisis support service Lifeline is 13 11 14. Other international suicide helplines can be found at www.befrienders.org.