SNP President Mike Russell, who stepped in to replace outgoing Peter Murrell, said the voting process must go ahead regardless.
Mr Murrell resigned on Saturday after taking responsibility for misleading the media on party membership numbers.
All three candidates vying for the leadership of the SNP have pledged to reform its operations.
Mr Murrell, who is married to outgoing party leader Nicola Sturgeon, resigned with immediate effect on Saturday, saying he had become a distraction to the leadership race.
He had been set to face a vote of no confidence by the National Executive Committee (NEC) had he not stepped down, the BBC has been told.
His resignation came a day after that of Murray Foote, the SNP's head of media at Holyrood. He had previously described press reports about the membership numbers as "inaccurate" and "drivel".
However, an NEC source told the BBC Mr Foote had been "thrown under the bus" by Mr Murrell.
The party this week confirmed there had been a big drop in membership numbers.
Speaking on BBC Scotland's The Sunday Show, Mr Russell said the SNP was "basically a good party" but things had gone "spectacularly wrong in recent weeks".
He said: "I think it is fair to say there is a tremendous mess and we have to clear it up, and that's the task I'm trying to take on in the short term."
Mr Russell said the most important thing now was that "they have a fair electoral process that produces a clear accepted outcome".
"We have got to have this concluded in the next eight days and then... the new leader has to look at the party and say let's rebuild this and let's rebuild the trust of Scotland."
He added: "This has not been an edifying process. There hasn't been a contested leadership in the SNP for 19 years, and it shows."
Mr Russell said he did not know that the SNP membership numbers had dropped by about a third over about two years, and said he did not know if it was related to the controversial Gender Recognition Reform bill.
"We were losing members and we were losing members that we should have known about, absolutely," he said.
"We were clearly not told about that. That is something I want to know why that took place, but I don't want to know it this week.
"What I want to know this week is we have got a process we can complete and can get a new leader of the party."
Two of the leadership candidates, Kate Forbes and Ash Regan, last week questioned the independence of the election process.
On Sunday, Ms Forbes said future decisions had to be made by "a big team" rather than a few people.
Speaking on Sunday with Laura Kuenssberg, she said SNP members wanted to know that the institution was democratic.
"There have been questions around, for example, the membership numbers that we've been looking for answers to," she said.
"I think at the heart of this is that the decisions within the SNP have been taken by too few people and I think that's well recognised across the political domain."
She added that members felt disempowered from the process.
"I think within government we need to make sure that it's a wide tent with a big team, rather than a very few people making decisions," Ms Forbes said.
She said she favoured a different approach to leading the party.
"We have a self-professed continuity candidate who says that he's going to keep doing the same things, and my response to that would be, you do the same things and get the same results," Ms Forbes said
"Let's put integrity, honesty at the heart, let's make the case for change and it's not just a change in terms of our policies, it's a change in terms of delivery and the culture of transparency," she said.
Leadership candidate Humza Yousaf has promised an internal shake-up from day one, telling the BBC: "There needs to be internal reform within our headquarters, of that there is simply no doubt.
"I've said from day one, since I launched my campaign, that internal reform is very much needed and certainly I will be looking to see what I can do to shake up that operation at headquarters from day one."
But during a visit to Glasgow Gurdwara, Mr Yousaf said the party's row over membership numbers was an "own goal".
Discussing the party's finances, he added: "I've not delved into this - I don't know the finances of the party because I don't hold an office-bearer position. But clearly If I'm elected leader of the SNP it's one of the first things I'd want to get up to speed on."
Ash Regan said she was "encouraged to see the democratic foundations of the party now asserting their rightful function".
She added that the SNP's foundations were based on accountability, transparency, modernity and accessibility.
Ms Forbes also told Laura Kuenssberg that on approaching the contest, she had weighed up having a young family with future potential responsibilities.
She gave birth to her daughter in August last year and was on maternity leave when First Minister Nicola Sturgeon announced her resignation.
"I've had to juggle a very young family with the contest but many mothers do that," she said.
Ms Forbes added that the role of first minister should be open to anyone, irrespective of their personal circumstances.
Within a couple of days of confirming she would stand in the contest, Ms Forbes found herself at the centre of a political storm.
She lost several supporters after telling journalists she would not have voted for gay marriage had she been an MSP at the time.
Subsequently Ms Forbes told Sky News that she believed having children outside marriage was "wrong" according to her faith as a member of the evangelical Free Church of Scotland, while stressing that: "In a free society you can do what you want."
But on Sunday with Laura Kuenssberg, she addressed the gay community, saying: "I give you an honest pledge today to govern in a way that delivers for you, that does not in any way undermine your rights and actually seeks to enhance your opportunities in Scotland to ensure that Scotland is truly that tolerant and pluralistic nation that we all want to see."
Ms Forbes said that people were "rightly scrutinising" who she was and what she believed.
She added: "I also think that in a pluralistic society, in a tolerant society, we can find a way to live together and to defend one another's rights.
"It's incumbent on me as a person of a minority community, somebody of faith to defend other minorities' rights and I hope that they might defend my rights too, that is truly the definition of a tolerant society."
Ms Forbes also confirmed that despite her faith, she would work on Sundays if she was appointed first minister.
She said: "The nature of the job is, of course, that it's 24/7. I recognise that. I hope nobody would begrudge me some hours off every week to be with family because I think that certainly makes for a more balanced life and hopefully better decision-making."
Opposition parties have criticised the SNP's handling of the leadership race and suggested Mr Russell was not being honest about party membership numbers.
The Scottish Conservatives said Mr Russell must have been the most "hands-off party president ever" if he did not know about falling numbers.
Party chairman Craig Hoy said: "He's asking us to believe that he had no idea what the SNP membership figures were until they were published a few days ago, nor who was responsible for ordering the party's former chief spin doctor to rubbish a perfectly accurate newspaper report on that figure.
"This is symptomatic of the secrecy and lack of accountability which infects the top of the SNP."
Scottish Labour deputy leader Jackie Baillie said the "election is in chaos" with allegations and resignations appearing on a daily basis.
"There is no way the president of the SNP did not know about the exodus of members from the party," she said. "Mr Russell should 'grasp the thistle' by facing up to the chaos in his party.
"If this is how the SNP run their own party, just imagine the chaos in government - Scotland deserves better."