Also, you are just wrong about the big names you mentioned:

Europcar

Debit Cards – Accepted in many countries, but restrictions apply (see below)

In some countries, only credit cards are accepted—unless you've prepaid online with a debit card. If you prepaid your rental with a debit card, you must bring the same card to the counter, along with a valid credit card for the deposit. Your card must be in the name of the main driver, and valid for the entire duration of the rental.

AVIS

Mini and Economy class; debit or credit card;

FVMR; debit or credit card;

Passenger van (9 pers.); debit or credit card;

Van; cash, debit card, or credit card;

Hertz

Your debit card is now welcome at Hertz in Europe. Here at Hertz we like to make renting a car personal. You decide what to drive and where to drive, and now you can choose how to pay.

We welcome debit cards across our European locations. We want your journey with us to be easy, so giving you options on how to pay puts you in the driving seat. No complicated processes – a simple deposit that works the same for credit cards as debit cards. Go your own way. Pay your own way.