I have many examples of this, but let’s focus on one:
Johnny Cash - A Boy Named Sue
from the album The Best of Johnny Cash
This song doesn’t have a cover, but it should.
A bit of information:
897b1a9b-88a2-38ab-8cc7-93f8932b1348
Probably your tagging settings at the time?
The image loads fine for me, in Picard:
P.S. That image is incorrect for that release, I am replacing it now - if later pictures differ, that is why.
Which settings should I give you?
ok that looks ok - so now reload the files in Picard and see if it pulls through the artwork like in Aerozol’s screenshot
Looks to be loading, save the file and it should now embed the artwork
What about other cases? I have over 1000 of them. 
Load and save them the same?
Make sure you have “embed cover images” or “save cover images as files” active, otherwise Picard won’t load the images
As you are after generic artwork… also look at some of the artwork plugins
@outsidecontext how many times can I load my 14,000 files into Picard??
After all, the current state without 1,200 covers is after running my entire collection through Picard. And not once.
As many times as it takes for you to work out your workflow and your settings
@aerozol you don’t understand.
I’ve loaded my music into Picard many times with different slider settings.
The result is always 1200-1500 songs without covers.
What else should I write?
As this thread shows, if MusicBrainz has the cover + the Picard settings are set correctly = your files get a cover.
Why your files don’t have a cover is not a question other people can answer. All we can do is confirm that Picard and MusicBrainz is working as expected. If you have matches where a cover should be loading/saving, but it isn’t, you can give us the specific link and we can take a look 
You don’t need to load your entire collection into Picard over and over again. That’s your choice. I would tag the files once properly, make sure they are matched against the proper release. Then you can load them into Picard for updating the tags if needed.
But we had this discussion before, and in the end it’s your choice.
Not all releases have cover art on MB. I can’t tell you generically why the files did not get cover art. You’ll need to look into the individual cases.
What about you save the 12500 files that got cover art, then go through the rest?
There are multiple options. You can choose a different release, upload missing cover art, or use some additional cover art plugins (such as TheAudioDB or FanartTV).
