Site-built deck drainage
Here is an article in Fine-homebuilding on site-built deck drainage.
In this article they use EPDM rubber roofing material.
I wouldn't hesitate to use this or creative alternatives, but I wouldn't consider this as a "waterproof/roof" system. I would consider it rain-proof where it diverts rain and rain-runoff from underneath the deck.
Retail deck drainage
There are products available that are designed for this as well. Like Timbertech Dryspace for example.
My opinion
Using either a site-built or retail product, I still wouldn't store anything under there that would be ruined from getting a little bit of moisture now and then.
If you want a real waterproof roof, then build it as a code-approved roof with code-approved materials installed how they were intended to be installed.
Issues
One potential issue I can think of right off the bat is that dirt and debris will collect in there, so do you have a plan on cleaning this out? You will need access from time to time to at least stick a hose at the highest part to rinse it out.
Another potential issue is that this would limit the airflow and potentially increase the humidity between the decking and drainage material. This could contribute to things like cupping or premature finish failure on wood decking.
Real Roof (info as requested in comments)
60mil vinyl can be considered an approved roofing surface. You would use something like this if you are not putting another surface over the deck.
Another roofing option (if you wanted to add wood planks) would be to do flat roof (i prefer a torch-down roof): then build panels out of your deck surface, or attach the decking to sleepers (see image below), or use wood tiles
I agree with Ecnerwal entirely. But, absolutely screw the deck boards down & don't use nails, unless you like them popping up & ripping your foot or catching a shoe or high heel. Screwing the boards down is structural, for mostly just the boards...it spreads the stress instead of snapping a board from a fatboy standing between joists.
Definitely, use the tar-paper & bravo on doing it in the past...almost no-one around me does it & laughs at me when I do it or point out it's missing. Their decks last maaaybeee 15-years & mine last easily 30+.
Go with stainless steel screws! Square-Drive like the old screws is fine, but mostly Star-Head is all I find anymore. I think chamfering the bottoms will only trap more debris on top of the tar-papered joists, this will also cause water to travel to the meat area rather than just the edge. Simply hosing, brooming or putty knifing out the new gaps annually should be greatly successful.
Best Answer
Left the barbecue on too long?
Of course the middle deck boards need to go, but you have the option of splicing the joist.
What you need is a 6 foot piece of the same material, screws and some good construction glue - the kind that comes in a caulking tube, not a squeeze bottle. PL-400 is good stuff.
Cut the new board into 2 x 3 foot lengths, apply glue generously, and sandwich the damaged section between them. Screw in from each side. No need for many or massive screws - that's what the glue is for.
Optional: chisel out the burnt section - makes the smell of freshly burnt deck go away faster. Paint the newly-exposed wood so it doesn't absorb water.
It may be possible to do all of this from underneath the deck, so you won't need to remove more than the damaged boards.