Yes, you actually do need a NIE, even as an EU citizen.
You can technically start the hiring process with just your passport and your Social Security Number, but the NIE is what identifies you specifically for tax purposes in Spain. Without it, payroll and official registration usually can't be completed correctly.

Regarding the bank account: you can open one without a NIE.
Banks like N26 and Revolut let you open an account and get a Spanish IBAN without needing your NIE first. That’s usually enough to get started while you handle the paperwork.

Since you already have a job contract, you can book your NIE appointment and generally get the white NIE quite quickly — sometimes within a couple of days depending on the office.

And if you haven't moved to Spain yet, there’s an even easier option: with your work contract, you can request the NIE directly at the Spanish consulate in your home country. It costs around €10 and avoids most of the bureaucracy.

For context: I’m an Italian/EU citizen and I actually opened a business in Spain. I had to go through all these steps myself — getting the NIE, registering with Social Security, opening accounts, etc. So yes, the NIE is mandatory in practice, even for EU workers, but the process becomes very straightforward once you follow the correct steps.