My next thought is to blanch the cherry tomatoes whole, then do a quick run in the food processor to break them open (but not crush the seeds - which I think the blender did last time), and then push the result through the strainer (before cooking down).
Who knew this part was going to be a project… 
Re bitter tomato sauce / purée / passata:
I tried the baking soda method, adding butter, and so on. Didn’t help much (and I can always taste baking soda so there’s that). I did strain out the seeds (and skin) finally.
I used a portion for an Indian chicken curry with plenty of spices, and cooked long - that worked well and the bitterness faded into the background.
Then we tried using it in a vegetarian curry which wasn’t cooked as long, and the bitterness was still very prevalent.
So - heavily spiced and long cooked seems to be the only salvage use (so far) for the bitter batch.
As Monk would say, having a lot is both a blessing and a curse. You can afford to “waste” the less than ideal parts, which for me is the skin and seeds. Composting makes it more acceptable, but I have also used the skin for tomato powder and “salt”.
Or just throw skins and seed out . I know its hard, but let it go! That’s in some Disney movie, right? If it’s a Disney movie, it must be okay.
Since you are cooking them anyway - I wonder if you could stick them in a hot oven for a while so the skins split and the flesh starts breaking down a little - then strain - then cook down to a real sauce consistency.
I’d still try to strain before cooking but cherry tomatoes add a layer of complication., so might ease that a little.
Update. I bought the juicer attachment to the big FP, which apparently does a great job of separating skins and seeds. Waiting for it to arrive.
In the meantime, what’s interesting (to me, probably not anyone else 
) - well a few things.
The cherries have A LOT more seeds and water than the other tomatoes (I have mostly cherry tomatoes growing). There was no bitterness when I cooked with the fresh tomatoes on a few occasions. So the vitamix was a bad idea, clearly, because it must have crushed the seeds. The FP did fine breaking them down without bitterness, as did the immersion blender.
So now I’m wondering whether to just run some of the tomatoes through the FP (without juicer) and leave them chunky, vs the passata that the juicer will result in. But there’s a part of me that’s a bit worried that cooking down the chunky sauce may still extract bitterness. An experiment may be warranted.
I can’t really visualize a puree of cherry tomatoes being successful. I’ve seen some sauces that are referred to as “popped” or blistered, or something to that effect. They go in the pan whole. Are you going to “can” them so they are shelf stable, or freeze?
I think roasting is the way to go with cherries. The one I tried, called a confit, turned o u t well, but specifically said not suitable for freezing. This one sounds decent, but I haven’t tried it.
You’re my hero- what a chore to try to seed cherry tomatoes. I live in southern AZ and cherry tomatoes just grow better here than the larger ones, which probably translates to less surface area for the sun to wreak havoc on. But they’re a bear to make things with if you don’t like the seeds, I really don’t mind them, or the skins, but I think the blender was your bad guy there.
Two little bits of random information that were in my head:
One of my favourite commercial tomato products has seeds and bits of skins in it. The ingredients list is “Tomatoes, salt”. I guess their processing method just doesn’t crush the seeds or something, but I have a hard time getting any sauce made because I just eat the stuff out of the jar.
Second thing: I went into a food store that specializes in Russian and eastern European goods, and they had jars upon jars of pickled whole tomatoes prominently displayed. So if in doubt over pickling, it may be good to check with some Russians. 
It seems nearly time to start seeding tomatoes here, in the PNW. So… thoughts on preserving cherry tomatoes.
I roast whole, on very high heat; basically charring the skins. Pull the skins off, saving for stock. Pack the tomatoes into small jars, with citric acid (see the National Center for Home Food Preservation website for details), and process in a water bath, 20-40 min, depending upon the size of the jar.
Sometimes I hot-pack unroasted cherry tomatoes, into 1/2 pint/250 ml jars, which are perfect for a “fresh” tomato/olive/artichoke pasta for 2.
Cherry tomatoes make a great tomato jam. I peel and seed for that. Blanch the tomato, pull off the skin; cut along the equator, and squeeze the seeds/pulp into a fine mesh sieve over a jar. The resulting liquid can be used for tomato martinis, to add to gazpacho, or any other recipe you’d use tomato juice.
