Bob and Judy nailed it. You need to get a second opinion and three estimates.
In your first post you said;
We have an old gate opening they want to remove the old one and install cleanout on outlet side of sewer check valve.
But in your last one you said;
we do not have anythinig no devices no check valve. ]
So what's this "gate opening" that you speak of?
Do you have a basement? If so is the sewer under the floor or does it exit above ? Tell me more about your installation! The way I figure is that if you can flush a toilet and have the discharge drain to the sewer and then you pull that toilet to snake the drain there should be no reason that they can't snake the sewer unless there's something installed that won't let the snake pas. Something's not right here. Perhaps you need to run a sewer cam down the line and see what you really have out there. Good luck, Tom