The authority also elected its first ever female chairman, as Councillor Ann Hartley replaced Councillor Pate in that position.
Councillor Pate, who represents Albrighton, described his appointment as a "tremendous honour".
And he was urged by fellow councillors to be "open and transparent".
Councillor Roger Evans, leader of the Liberal Democrats on the council, said: "I hope members of the public are treated with respect when they petition or speak at meetings.
"I would urge you to use your office to regain the respect this council deserves from the public."
Councillor Pate was also asked whether he would look at, what Councillor Evans described as a "failing" scrutiny process.
The calls for transparency were echoed by Councillor Alan Mosley, who also demanded a need to restore ip&e services to "proper, local control".
He said he wanted to see a commitment to "local government and services which are being diminished".
Councillor Pate said: "Council tax has been the same for the past six years and we have the new university in Shrewsbury which attracts the young people the county badly needs.
"We can't disregard what's happened over the last decade but the times they are a changing. They must change if we are to deliver sensible local government to the county."
Councillor Pate, who was leader of Shropshire County Council for six years up until 2009, was elected by Shropshire Council's Conservative group to replace council leader Keith Barrow, who stepped down a fortnight ago.
He acknowledged the problems the authority faces today are different to those the old county council faced under his leadership, and pledged to work with the county's MPs to ensure Shropshire gets a better deal.
He added: "The staff are the most important resource and asset this council has. I want them to know they are appreciated.
"Hopefully we can all work together to improve this authority."
Meanwhile, Councillor Brian Williams, who nominated Councillor Hartley, member for Ellesmere, to be chairman, praised her for the work she has done for the council over the years.
He said: "I have long been an admirer of Ann Hartley. She was a highly effective cabinet member and a footsoldier. She's also been a highly effective mayor of Ellesmere.
"She's the first lady chairman of this authority."
Councillor Hartley thanked fellow councillors for "bestowing this huge honour" on her. She said: "I would like to thank Malcolm Pate for his tenure. My job now is to bring back some civic pride to Shropshire.
"We're elected to represent the community in our beautiful county, we need to be proud of that."
Councillor Hartley also praised "amazing" council staff for their hard work and loyalty.
Meanwhile, Shropshire Council's private sector housing team has been announced as a finalist in two categories in the local government chronicle awards 2016. The team was shortlisted from over 120 entries.
Councillor Lee Chapman, Shropshire Council's cabinet member for adult services, said: "We are delighted to be a finalist in these nationally-respected awards. Our partnership approach between the council, private sector and voluntary sector agencies is delivering outstanding results."