Tomato paste is just tomato puree/sauce that has been cooked for a long time and reduced, and then reduced some more, and then reduced and concentrated some more, and then some more. 2 14.5 oz cans of tomato paste is the tomato-ey equivalent of 6 cans of tomato puree! It is not a thickening agent - it is just thicker because it has had most of the water cooked out of it.
For a less tomato-ey chili use less tomato paste (about 1 8 oz can) or 1 14.5 oz can tomato puree/sauce. To thicken, use a slurry of masa harina or AP flour or corn starch. Masa harina is the authentic thickener - AP flour or corn starch will work but have a different flavor.
I only add diced tomatoes if I am making "chunky chili" for some specific purpose - generally the only lumps in my chili are chunks of meat.
Chili powders vary - and yes they are a blend of ground chilies and usually cumin, Mexican oregano, salt and may include onion and or garlic powder/salt. What makes different chili powders different is the blend and what chilies are used. For a more authentic San Antonio flavor look for Gebhardt's chili powder. Oh, not all chili powder is "knock your socks off" hot - some of the best isn't. If you want more heat add more and/or different chilies, hot paprika or cayenne pepper.
Phil - don't worry about using pork, I won't call the chili police on ya'!
A couple of nights ago my DIL used ground turkey!!!