Scott Morrison’s warning that the ABC board could “expect a bit more attention from me” if it did not “do better” has prompted a key Senate crossbencher to suggest stronger legislation to safeguard the national broadcaster from political interference.

The prime minister used his Sunday appearance on ABC TV to put the organisation’s board on notice, despite allegations of political interference having led to the resignation of the chairman, Justin Milne, in a week of turmoil that included the sacking of the former managing director, Michelle Guthrie.

Tim Storer has called for an overhaul of how the board positions and the managing director are selected, saying he was “troubled” by Morrison’s comments, given the events of the past week.

“The position of chairman and managing director should not be filled permanently unless and until new legislative arrangements are in place to strengthen that independence and transparency,” Storer said. “Several directors have been appointed by the current government without following the panel’s recommendations, undermining the value and credibility of the panel.”

Storer suggested legislative change, advocating for the establishment of a US-style independent panel that would publicly publish its recommendations and force the government to publicly explain why it was choosing to ignore those recommendations, if making its own appointment.

Documents obtained by Guardian Australia, first published last week, show the five most recent appointments to the ABC were directly made by Mitch Fifield, with the panel’s recommendations either circumvented or ignored.

Storer said Morrison’s weekend comments had concerned him to the point he now believed the parliament needed to act “urgently” to “guarantee the independence of the ABC”.

The opposition has also called for urgent change, with the Labor leader, Bill Shorten, calling for any future permanent appointments to be bipartisan, “given these publicly revealed attempts to undermine and politicise the ABC by the current government”.

“Labor is incredibly concerned about the ability of Minister Fifield to make suitable appointments, including for the new ABC chair, given his disregard for the independent nomination panel process,” Shorten wrote in a letter to Morrison. “This approach cannot continue. The government’s partisan approach to the ABC is putting its independence and integrity at risk.”

Morrison did not explain what form his “attention” would take if the board did not improve to his standard, given the ABC’s legislative independence from the executive government, and on Monday would only reiterate his demand the broadcaster “get back to work”.

“As I said on the weekend, the ABC have got to stop talking about themselves and get back to work and do their job,” he said from Perth.

“That’s what people expect of them and, when it comes to the issues which are really of burning interests to Australians, it’s the drought, it’s jobs, it’s electricity prices. It’s all of these things, that’s what they’re focused on, and, you know, the ABC is a very important institution but frankly there are a lot of issues out there that I think are really burning through kitchen tables as topics of conversation more than that one.”

The ABC continued broadcasting as usual as the turmoil played out at board level. Morrison made no comment about the government’s most recent decision to cancel parliament during the Liberal leadership crisis that saw him become prime minister in August.

But he did rule out Storer’s overhaul, and also Labor’s suggestion that the next permanent appointments be bipartisan, by saying he saw no reason to change the process.

“There’s a process for appointing chairs and board members to the ABC,” he said. “We’ll follow that process. Elected governments make appointments to these boards. I’m sure if they have suggestions, they’ll make them. But the government runs that process.”

Last week, the government nominated board member Dr Kirsten Ferguson as the interim ABC chair. She vowed to maintain distance from the government to ensure the broadcaster’s independence was protected.

In his outgoing interview with the ABC, Milne said he believed the chair’s role had been to act as a “conduit” between the ABC and the government, not a “wall”, adding “you can’t go around irritating the person who’s going to give you funding again and again and again if it’s over matters of accuracy and impartiality”.