In a statement shared today, Andy Street said the authority "will not let this lie and have a plan in place".
It comes after the 258-year-old pub was demolished after a fire on the weekend.
Staffordshire Police have since confirmed it is treating the fire as arson, but no arrests have yet been made.
A giant rubble of bricks is all that remains of The Crooked House.
In a statement posted on social media, Andy Street said: "It is clear from our conversation with South Staffordshire District Council that they remain just as committed to the future of the Crooked House as we do, and we want to thank them for their time and candour.
"They will not let this lie and have a plan in place (including pursuing enforcement action as they announced earlier in the week), but they must be given space and time to conduct their work - just as is the case with Staffordshire Police and Staffordshire Fire & Rescue Service.
"That is why we would really encourage people to steer clear of the site now and allow authorities to carry out their work.
"We feel the sadness, anger, and frustration as much as anyone over what has happened to the Crooked House, but the last thing we want is for well-intentioned community action to inadvertently damage any positive future for the site.
"We have been very clear from the moment the Crooked House was demolished without permission that we believe it should be rebuilt brick-by-brick, and we remain laser-focused on making that happen.
"Nothing we've seen or heard this week has led us to change our mind.
"We are absolutely on the case, and indeed our resolve has hardened after hearing from the District Council - whose plans we have pledged our full support to.
"Whoever has targeted this beloved landmark in this way has messed with the wrong pub, the wrong community, and the wrong authorities.
"The Crooked House will not be consigned to history on our watch."
The pub burned down just two weeks after it was sold by pub company Marston’s to a private buyer.
The loss of the pub, built in the 18th century, prompted a wave of anger from politicians, local residents and former regulars.
It was destroyed just two weeks after it was sold by pub company Marston’s to a private buyer, which is understood to be ATE Farms Limited.
The company’s only active director, according to Companies House, is Carly Taylor, 34, with George Adam Taylor, 44, a director from 2008 to 2014.
Mr Taylor was previously a director of Himley Environmental Ltd, which owns the 15-hectare Oak Farm Quarry and Landfill site adjacent to the Crooked House and is registered to the same address as ATE Farms.
The site’s total demolition had not been sanctioned by South Staffordshire Council, which said on Tuesday that it had only permitted the top floor to be demolished over safety fears.
In a statement, it said it had not agreed to nor deemed necessary the demolition of the entire structure, which had become a popular local landmark and was dubbed “Britain’s wonkiest pub”.
It is currently exploring whether the demolition constitutes a breach of the law.
Following the meeting with Mr Street, councillor Roger Lees, leader of South Staffordshire Council, said: “We’d like to thank Andy Street and his team for their time in meeting with us yesterday and their full support for the approach being taken by South Staffordshire Council.
“Our officers continue to work to explore all options in relation to the building and planning breaches at the Crooked House.
“I want to assure all interested parties that this case is absolutely a key priority for our officers and we are committed to working in partnership with all relevant parties on this important matter.
“We urge people to stay away from the site and to allow the council, the police, and other agencies the space to focus on the important tasks in hand.”
Staffordshire Police said on Thursday that it had no control over the building’s demolition, but that it was engaging with its owners as part of enquiries.