Let me continue from my previous answer and try to address the issue about testing.
One thing to look at is simply adhesion of whatever paint you plan to apply on top. You mention working in oils, so the following link will not be as relevant, but we have actually had an ongoing issue of artists seeing adhesion failures on top of inexpensive preprimed supports, to the point that we wrote an article on the topic and shared some very simple testing protocols artists could use to see if their canvas might pose an issue (essentially just putting a drop of water on the surface and observing the contact angle):
Pre-primed Canvas Adhesion Testing Procedures
Clearly the failures speak to some form of contamination or incompatible material, and while we have not tested oils for the same issues, I would be concerned about using any canvas that could not pass this test as it sends up a red flag.
It appears that you know about the crosshatch adhesion test, but if not you can find that covered in the following piece under Adhesion testing:
We would recommend testing a few colors of different types, and to test after they are thoroughly dry and can pass the thumbnail test.
You should definitely test for oil strikethrough. A couple of ways to do this. One, which is quite a hard test, is to apply three drops of linseed oil to the canvas and let it sit. It should not form an oil spot on the opposite side and, equally important, should not dry to a hard bead sitting purely on top. It should slightly soak in - showing good wetting out and mechanical adhesion - but not so much as to be drawn through to the other side. Alternively, apply a generous layer of a slow drying oil paint to see if the oil penetrates through.
Another test is a simple fold test for flexibility, where you should be able to fold a sample of the primed canvas completely back onto itself 10 times without showing any signs of cracking.
Moving on from there, test for yellowing. this is a little harder for you then us, as we have accelerated UV exposure equipment, but you can certainly try to at least tape up a sample (while keeping another in dark storage) to a south facing window and check after 3 and 6 months, then again after say 1 and 2 years. It should remain white.
You can test the pH of the surface with a simple pH test pen (Google it) and you should get a neutral or alkaline reading.
Test under a UV black light for fluorescing - it should definitely NOT fluoresce, and any glow would indicate optical brighteners, which degrade.
Other types of tests would require a dedicated oven to expose sample to 140F for extended periods, which can give you a read on accelerated aging of the binder, and is something we do. Also if tehre are plasticizers in the coating - as many inexpensive vinyl ones will have - those will tend to evaporate away with heat and so embrittlement can often be seen with prolonged heat exposure.
There are other tests but let me stop there. As you can see there are a lot of aspects to look at, and some would be impractical for you to really test - such as flexibility at different temperatures and humidity levels.
Finally, let me leave you with one story about someone who used a very high quality commercial paint for many years as a ground for their paintings without having any problems. All of a sudden, however, our High Flow paints were beading up and would not adhere. They called us to see if we could figure things out. We went so far as to purchase the primer to try and reproduce the issue - and sure enough, we could. Looking into it, what we discovered was that the paint was now 'new and improved' and touting cutting edge stain resistance and easy wiping off of grime and dirt. All sounds great in a housepaint! However, to achieve this they were adding things into the paint to make it very hydrophobic once dry, thus the reason our paints would not adhere. Its a great case where even a high quality commercial product often is aiming for different properties and qualities then you need or want in an artists' material.
So, we now its a pain and not the cheapest solution, but we would actually counsel you to do whatever you can to afford and use the best quality artist grounds as you can as really the entire life of the painting is riding on it. Literally.
Senior Technical Specialist
Golden Artist Colors