model of motherboard - Vostro 3510

You are wrong.
For example, the Dell Vostro 3590 has a motherboard whose model is 0XHR1J.

You can check this for example via HW INFO.

I ask because these models have a manufacturing defect. The USB does not work well. With prolonged use it breaks the connection. This results in logical damage on the connected drive.

Therefore, I want to buy a laptop that does not have such a motherboard.

That isn't the model number.  It's the part number for the board.

If you want a system with a different type of board, you need to purchase a different model system.  Though there may be other part numbers available for your system, they will all have the same design.  The part number will change, for example, if you source a board with a different CPU or GPU -- but the board itself will be the same other than with regard to the installed components.

DELL service told my colleague that this is the model of motherboard in this laptop.
Which is confirmed by HW INFO.
So, will the Vostro 3510 have a different motherboard to the Vostro 3590?
I'm keen for the USB ports to be handled by a different chip to the Vostro 3590, where this is faulty.

Any model other than the one you have will use a different system board.  The boards are custom to the system in which they ship, and very, very rarely interchangeable between even outwardly similar models -- the mounting holes, dimensions, and port locations are commonly different.  It's also not uncommon for for different model number systems to have been designed and built by different contractors -- one may have been made by Wistron, and the next or prior generation by Compal, etc.

If you want assurance that the USB controller is different, you need a completely different model system than the one you have - there is no board that will fit your system and meet that criterion.

What do you mean when you write 'system'?
Is the Vostro 3510 a different system than the Vostro 3590?

They are very different systems.

Mainboards work only in the exact same model system -- i.e., ONLY a Vostro 3510 board will fit a Vostro 3510.  It will not fit any other system.