Why should you give them your bank routing and account numbers? So they can direct-deposit your pay.
Why not demand a live check as your pay? Hassle. You're making more work for the accounts payable person responsible for getting you paid. They have to receive the live check from their payroll company, find you, and hand it to you. They may have to cover the cost of express delivery and insurance for a package containing just your check.
Is direct deposit good for you? Very likely yes. The funds hit your bank account right on time (or maybe even early) and are generally available immediately: the bank trusts the payroll company so you don't have to wait for a check to clear.
Is it a routine request? Yes. In the US receiving pay by direct deposit is the norm.
Is it safe? Yes. Both your employer/customer and the payroll company have far more to lose in reputation and money than you do if they try stealing from you. They aren't going to do that. Police and judges take wage theft very seriously.
Here's something to think about: As a contractor / gig worker, it's important to be really nice to the person who processes your payments. They can help you with all sorts of stuff related to billing and payments. A smile can get you paid a couple of days sooner sometimes. A live-check hassle is the opposite of being nice.
With respect, I hope you reconsider your position. Direct deposit is normal and good. Seriously.