A big part of this style of sauce is the chunkiness. We used to make tons of these types of sauces growing up, and some things changed the flavor more than others. Leaving it a bit chunky is best for this style (with carrots/etc), i.m.o.

Straining it changes the flavor the most, in my experience. It filters out the pepper skins, seeds, carrot bits, etc which add a significant portion of the flavor. I find it more astringent and less sweet after straining. At that point, I'd prefer to make "sport peppers" (fermented peppers and garlic left whole while the vinegar solution is used as a sauce).

Blending it to a fine paste seems to have less of an effect on the flavor, but it makes it a lot hotter. I'd assume that's because it releases more of the capsicum in the seeds. It also removes the variability you get with a chunky sauce (e.g. sweet bit, then hot bit). Whether that's good or bad is a matter of opinion, but I rather like the variability.

Hmm... And on a side note, I just realized that the rest of the world does not use the term "sport peppers" the way I do (the exception being a couple of regional brands). I never thought of them as the peppers on a Chicago style hot dog... Today I learned!

You could probably hit it with some kind of a blender if it's too chunky.