How can I make onion tomato paste taste more punchy?

Raw is punchy!

Reserve some of your crushed garlic and onion mix. Mince it fine. Then add it at the end. Cooking brings out some allium flavors and attenuates others. If your figure out the raw punch is what you are looking for, experiment with the ratio of cooked to raw, or experiment with just raw onions or just raw garlic.

Or you could consider bringing in different members of onion / garlic family which could also give your mix different depth. For example some people find raw green onions or raw sweet Vidalia onions more palatable than raw yellow onions / garlic. Shallots are similar to garlic but less pungent and if raw garlic is too much raw shallots might work.

On the tomato end I agree with @myklbykl as regards additions. Worcestershire sauce is one I use. Tomato paste in a small quantity can add more tomato punch. Anchovy paste is available in tubes and can be good for this sort of thing.

A. To make the flavor of tomatoes stronger in a sauce:

1. start with better-tasting tomatoes. Depending on time of year these will often be canned. There are many reviews of canned tomatoes, so I'll not pass judgment here but there are some excellent ones out there.
2. reduce the sauce more to intensify the flavor.
3. Tomatoes are an umami flavor. Adding another umami to it could intensify the flavor you're looking for (such as marmite, anchovies, etc.).
4. if you want to get really fancy you can try playing around with a centrifuge such as a Spinzall.

B. To make onions and garlic stronger you can try combining sweated fresh onions and garlic with powder. Be sure to activate the powder in water before cooking directly in fat or the flavor enzyme will be deactivated before it can go to work.

there's another approach altogether that has worked for me: lacto-fermented tomato sauce. only takes a week, triples the amount of dishwashing, and could easily go completely wrong, especially on the first try!

• same amount of raw tomatoes you're already starting with, quartered, and keep the juice!
• halve the amount of onions you'd normally use, because the flavor doesn't get softened by heat, diced and puréed
• (garlic, puréed. I add half the weight of onions used, but that's a lot of garlic)
• 1/2 tsp salt per pound of tomatoes (5.1 g salt per kilo of tomatoes)
• 1 tsp cultured buttermilk per pound of tomatoes used (about 10.2 g per kilo)

pick a vegetable fermentation guide on the internet, follow the process, but with these ingredients, mixed well and mashed down very well to eliminate bubbles. if all goes well, in 7-10 days, you can purée the results and pour your sauce over pasta or whatever. it tastes fantastic, fresh, bright, and hopefully very punchy.

This sauce can be cooked, but fresh flavor will be lost.

If you have questions about lacto-fermented tomato sauce, you should definitely create new questions, as this has veered far from the original topic, and I honestly don't know how well this answer will be received by the community.

more options:

1. add more tomatoes via pre-concentrated products like tomato paste or powder
2. when initially cooking, add some rinsed tomato stems and leaves, this should help with the "punchy" part, just be sure to remove them soon after cooking
3. after cooking, add up to 1 tsp lemon juice per medium/large tomato. if balanced well, the lemon flavor will lift the tomato flavor without actually tasting lemony
4. other spices like fresh-ground black pepper may give you some of the desired flavor punch

cooking time might matter, too. a long-cooked marina tastes very different from a quick sauce of mashed, fresh tomatoes