Rent a wet and dry sofa/carpet vacuum, like a Kärcher Puzzi (I know). I did recently for my sofa - a good machine will cost 1500 or so, but I only paid 30 to rent it for a day.
They’re dead easy to use and you’ll be shocked at the amount of grime that comes out. There are tutorials on YouTube, here’s a good one.
Baking soda, vinegar, and and all those other house-wife hacks are bullshit and won’t help you at all.
As an aside; baking soda and vinegar does fuck all. They break down to salt water. There is no cleaning magic happening. You can use baking soda or vinegar depending on what you clean, but most certainly not on a mattress that requires deep cleaning.
Finally invest in a protector and possibly a waterproof pad in between. The protectors are washable, so chuck that in the washer every so often and you’re golden.
My tiniest caveat is baking soda and vinegar is meant to create microscopic bubbles that can physically lift and agitate things -- but you are correct it's not a chemical/solution for cleaning - just a little micro scrubbing for particular use cases. .
Use a good steam cleaner and enzymatic cleaner that is often used for biological messes like pet urine. The enzymes will break down the odor and remove it.
Don't mix chemicals together. that can be dangerous.
If you Google "clean mattress with baking soda" you'll find a bunch of similar recipes. Lots suggest using essential oils, which I didn't use.
The results were decent when I did it. Definitely de-funkified the mattress a bit, and removed some of the stains. It was hardly like new, but it was definitely better than before.
Bathe before sleeping everyday. It not only keeps your sheets clean but help you sleep well too. To clean the mattress, use those steam vacuum cleaners and then air your mattress in the sun if possible.
Probably a bit crazy, but I used a carpet washer on my bare mattress with all the covers taken off. The zipped on covers were easily removed and put in the washing machine. Just hoping to extend the life of the mattress a few more years because they are so damn expensive.