Love the OXO mill as well! I make my sauce a variety of ways depending on quantity of tomatoes on hand and time. Cook either stove top or oven roast.
No matter what method I use I always remove seeds with the mill. As for the skins if I make sauce on the stove too I put the tomatoes in the Vitamix for a short time to grind up the skins. There are still a lot of seeds left so I pour the liquid through the OXO to get rid of the seeds and cook the liquid down.
Alternatively if I oven roast I cut up the tomatoes into a ceramic roasting dish, cook down and put the mixture through the mill to remove skins and seeds. I like Bower’s method of putting cooled cooked mixture in blender to grind up skins. But then I would use the mill to remove any seeds that didn’t grind up. Might change consistency but worth a try. Oven roasting seems to take longer than stove top but I prefer consistency and intense flavor.
A third method I have used only once was to cut up tomatoes, place on a tray in freezer, and transfer frozen pieces to gallon bags to store in freezer. I was visiting family in another state for a week and didn’t have time to cook the tomatoes before I left. Upon my return I defrosted the tomatoes. The skin came right off the pieces. I put the defrosted pieces through the OXO which worked because they were super mushy from being frozen. This removed the seeds. I then oven roasted the pulp that came through the mill. That went fast and I loved the consistency.
In all cases I add a little honey, salt, pepper, herbs, mashed garlic and Parmesan cheese to final product and freeze in containers.
Basically I find the OXO mill to be indispensable for making tomato sauce regardless of the method!

If you like refried beans the strainer in question makes one of the best around.
I used something more simple but if you have one why not use it.
The object is to mash up the beans after cooking in lardcand their juices then run through a strainer to get rid of the skins.
I'm doing canned kidney beans this way today.
In my opinion there isn't anything good about kidney bean skins.
Almost impossible to digest.