Yes, Experian, TransUnion, and Equifax all offer tenant screening services wherein the applicant does not need to supply the landlord with their Social Security number.
I use TransUnion's SmartMove for tenant screening, and I do not see SSN's of prospective tenants or account numbers on their full credit report. I would imagine other competing services are comparable.
A landlord may not be inclined to use such a service because without your SSN they are unable to use a debt collection service to collect unpaid rent. This is a relatively small risk for landlords, but it is a downside to not collecting SSN.
From what I've seen these tenant screening services don't do full income verification, but they do assess reasonableness of the applicant's self-reported income. That may or may not be good enough for some landlords. Bank balances do help paint a picture of you as an applicant, but landlords don't need to see account numbers (though if you pay by check they'll see that one anyway).
A Driver's License is many people's only form of photo ID and therefore typically copied for identification purposes, or at least used to confirm the major pieces: Name, DOB, Address.
There are good arguments in favor of the tenant screening services and withholding SSN, but I think they're better suited for small-time landlords. Large companies often do sufficient volume of background/credit checks that they likely have cheaper arrangements and like generating revenue from their application process. Don't be shocked if they stick with their requirements.
Edit: As far as how the process gets initiated, the landlord creates an account, then they initiate the request. With TransUnion's Smart Move I create an application for each listing, and when I want to move forward with an applicant I provide the applicant's email, then the applicant gets an email from Smart Move stating that I'm requesting a background/credit check related to the specific listing and instructions/link to proceed if they so choose.