Mariu Chrzaszcz:
I am going to the USA and want to hire the car for going on a tour of
states: TX, NM, AZ, UT, NV, CO.

In American English you want to *rent* a car.

The car will have the full cover but for one driver only (.

Try a different rental company.

Why are rental companies demanding payments for the additional driver
(wife)?

When I have rented cars as a tourist in the US, my wife has always
been allowed to drive at no charge. However, I have not done this
in the last few years. Maybe practices are changing. The rental
companies, like the airlines, are always looking for ways to add new
charges for things that used to be included in the advertised price.

Why? Because they can make the advertised price look cheaper.

During a road check/control can police demand holding such payments?

They can ask to see your registration papers, which means the rental
documents. If your wife is driving and you are the renter, I would
not expect them to worry about that, but they certainly could check
the documents and you could be in trouble with the rental company.
The worst case is if you have an accident and your insurance is voided.
--
Mark Brader, Toronto "As for Canada's lack of mystique,
it is not unique." -- Mark Leeper

My text in this article is in the public domain.