“The talks were very open, and from the part of those affected, no holds barred, and pointed to the weakness of our country in this situation,” Merkel said, as Germany held a day of solemn commemoration for the victims on Tuesday.

“Today must be not only day of sadness, but also a day of our will to make better things that did not work well,” she vowed, adding that she had offered to meet the bereaved again in a few months’ time.

President Frank-Walter Steinmeier also told the bereaved and emergency workers at a private church memorial for the victims that “it is true that some support came late and remained unsatisfactory”.

“Many family members and injured – many of you – felt abandoned by the state,” he said, recalling the words of a mother who had lost her daughter and said no one had comforted her after the attack.

“I can’t get those words out of my head,” he said, saying that the relatives’ appeal to be heard had “triggered something and set it in motion.”

In the hours following the assault, which was claimed by Islamic State, politicians had put on a brave front and repeated the mantra that Germany would not be cowed by terror.

But Steinmeier said such rhetoric had done little for the victims.

“So soon after the attack … these words don’t sound simply defiant and self-confident, but also strangely cold and detached,” he said.

At midday, Merkel joined relatives in inaugurating a memorial – a 14-metre (46-foot) golden crack in the ground engraved with the victims’ names.

An ecumenical prayer was held later to remember the victims, while at 8.02pm (3.02am Hong Kong time) – the exact time when Amri rammed his truck into the crowded square – the church’s bells chimed for 12 minutes.

But the run-up to the commemoration has been marred by criticism of the authorities by families of the victims.

A wrenching open letter by some of the bereaved accused Merkel of failing to personally offer condolences.

“They have not even tried to listen to us, to understand our needs,” said Bild Yaron, the brother-in-law of Dalia Elyakim, an Israeli woman killed in the attack.

Susanne Covington, 62, who came to light a candle at Breitscheidplatz, said Merkel “should have done something a little earlier”.

Justice Minister Heiko Maas apologised in an editorial in the Tagesspiegel newspaper: “We were not sufficiently prepared to deal with the consequences of such a terror attack.

“For that, we can only apologise to the victims and their surviving relatives.”

Kurt Beck, who was commissioned by the government to look into the handling of the aftermath, last week outlined a litany of official failings, including taking up to three days to tell anxious relatives that their loved ones had perished and even sending the bereaved autopsy bills complete with late-payment warnings.

Further adding to the embarrassment, the Bild newspaper said the letter inviting relatives to Tuesday’s commemoration was accompanied by an information sheet stating that “taxi costs will not be reimbursed!” and urging them to use public transport instead.

A spokeswoman for Berlin authorities, Claudia Suender, told Tagesspiegel that such information was “required under budgetary and administrative law” even if she “regretted the impression it gave.”

Police also faced fierce criticism after it emerged that Amri, who arrived in Germany in 2015 and registered under several different identities, should have been deported.

On Sunday, the Welt am Sonntag newspaper said the Tunisian had been under closer surveillance by Germany’s secret service than previously thought, suggesting the authorities might have left him free to detect his instigators.

Amri himself was shot and killed four days after the attack by police in Italy, where he had previously lived.