I think for many organizations, it's not the process (along the lines of what you outline) that has changed. It's the terminology and the underlying assumptions that have changed.
Rush ---> Recruitment
Pledge ----> New Member/Associate Member/Probationary Member/etc.
The newer terms often reflect a newer understanding that a "new member" is just like an initiated member except for the "initiated" part -- the new members are treated in most respects like initiated members. (Along with this in some orgs, there would be nothing like what you call a "bid to initiate." It is assumed that all new members will initiate.)
The one exception in the fraternity world I can think of is Sig Ep with its Balanced Man program.
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