There are a number of ways to implement the scheme with a number of different companies - or you can get your employers to run it themselves. It's not a lot of work, but it usually requires someone with a bit of enthusiasm to put a bit of effort in.
From your previous posts I know you're looking at a Boardman... the
Halfords scheme would therefore be the one you want.
Halfords do all the advertising etc and have a website that you log into to register and apply. It costs nothing apart from the time for an administrator to review and approve the applications, and the time it takes for payroll to action the salary sacrifices and your accounts team to keep track of the repayments.
The cheapest way is for your employers to fork out initially for the cost of the bikes. We have a 1-2 month window and then pay 1 invoice to Halfords for all the bikes bought in that period. Then we recover the money back from the employees over 12 months. Work gets all the money back, and they also make a fair bit of saving on employers NI. Staff save on income tax and NI so everyone's a winner

You're right that work have to buy the bikes, but as it's all recovered in 12 months it's no real loss to them. There are ways where your employers don't have to pay out anything, but that means taking out a loan, which obviously will cost those in the scheme as the interest repayments will come from your savings.
There's also the issue of the exchange of ownership once the loan period has finished. The easiest way to navigate this is for work to give the bikes to the staff as a benefit in kind, which incurs a small tax charge, but it's a lot cheaper than paying the HMRC guideline amounts.
Give me a shout if you need me to expain anything a bit more fully.