The education secretary, Nadhim Zahawi, has accepted the government committed errors in its attempt to save the former minister Owen Paterson from suspension, but denied the debacle called into question the prime minister’s judgment.

In a broadcast round, Zahawi accepted it was a “mistake” to try to link overhauling the standards system that had found Paterson guilty of paid advocacy with a parliamentary attempt to preventing him facing the punishment of suspension.

He admitted to not reading the standards committee’s report into Paterson’s conduct before voting in favour of overhauling the system.

Speaking to Sky News, he said: “The prime minister has always been very clear that paid lobbying is not allowed. The mistake is the conflation of creating a fairer system with the right of appeal for parliamentarians to be able to put forward an appeal process.

“Conflating that with the particular case of Owen Paterson was a mistake and I think the leader of the house, Jacob Rees-Mogg, came to the house yesterday … upon reflection, yes it was a mistake, and I think it was right to come back very quickly to the house and say we need to separate these things out.”

Paterson resigned as MP for North Shropshire after the government ditched its plan to set up a new standards panel.

On BBC Radio 4’s Today programme on Friday, Zahawi denied Johnson was trying to head off a potential investigation by the standards commissioner, Kathryn Stone, into the financing of the decoration of his Downing Street flat. Zahawi said it was “absolutely not true” that the new system that the government was forced to ditch was a “pre-emptive strike” against Stone.

He said: “That is absolutely not true, and Kathryn Stone and her duties are the responsibility of the House of Commons, and the Speaker of the House.”

Zahawi also defended his cabinet colleague Kwasi Kwarteng over his suggestion on Thursday that Stone should consider her position. Before the U-turn over the new committee to examine Paterson’s case, Kwarteng said Stone should “decide [on] her position”.

Zahawi told Sky: “I don’t think Kwasi was saying anything different that Kathryn Stone has a responsibility to the legislature.”

He added: “I think it’s important to remind all parliamentarians and the country as a whole that Kathryn Stone works for the house, for the legislature, and I think it’s only right that I echo the words of the Speaker by saying that it is up to the house how the commissioner and the procedures are delivered.”

He admitted on BBC Breakfast to not reading Stone’s report into Paterson, and appeared to take on trust his colleague’s claims of innocence. “Owen says that much of it is contested, right?” he said. Referring to supportive witness statements published in the report, Zahawi added: “I think something like 14 people have sent statements [saying] that it is contested.”

Zahawi reiterated the government’s determination to change the system, but with cross-party support. He said: “The important thing to remember is that parliament as the legislative chamber of our country has absolutely the right to look at and improve the system.”

Zahawi said the issue of the Downing Street flat had been looked at by Johnson’s own ministerial standards adviser, Lord Geidt, “and the prime minister was found not to have broken any ministerial code”.

He added: “I think it was looked at by Lord Geidt, it’s a ministerial declaration and I think that’s the correct way of doing this. We have very good robust processes. We always want to improve them, but I think that’s the correct way of doing it.”

He also denied that the government’s handling of Paterson’s case called into question Johnson’s judgment. “I think actually it says that the prime minister, when wanting to be following a process that makes the system fairer … wanted to do that.

“And very quickly realising that that’s one thing that we should pursue on a cross-party basis and we’ll come forward with proposals, and I hope, we can sort of set our politics aside and create a fairer system, because right of appeal, I think, is important, and your listeners will see that as important.”

Zahawi avoided questions about whether Paterson had breached the parliamentary code. Asked on Sky News whether he believed his colleague did anything wrong, Zahawi said: “The commissioner had investigated and had come back on the investigation around what Owen Paterson was doing in terms of his work for two companies.”

On the government’s U-turn, he said: “We have to take collective responsibility as parliamentarians. It’s a process that parliamentarians voted for.”