Cartridges do affect the sound a lot because they all have vastly differing frequency response. I don't think any of them are completely flat, but some of them are definitely less flat than others. I am pretty sure when reviewers talk about a cartridge sounding "lively", "bright", "airy", et cetera they are mostly talking about the frequency response (well that an probably a bunch of sighted bias too).
I've seen some graphs from Denon and Audio Technica that looked flatter than my speakers. When I had a DL-103 it actually came with a QC inspector print out graph of the FR.
But, yeah, a lot of the fan-favorite / cottage industry carts are pretty spiky.
Here are the FR graphs from
Miller Audio Research report for the Audio Technica AT33EV, which is the main cart I'm using now.
I don't know why they separate the low and the high frequencies into two separate graphs, or the significance of their vertical vs lateral tests.
The AT33EV is a mid-priced ($469) LOMC with
decent (but not amazing) specs, a fairly tank like build (as an aside, I'm a fan of the value for money that the big scale cart manufacturers like Audio Technica, Ortofon, and Denon provide when compared to boutique makers), a stylus guard (critical to me these days after the housecleaner killed earlier MCs), is the nth-generation of a design that's been around since the 1980s (I think), also comes in mono (I also own) and Shibata flavors, while the EV version comes with a workman-like alu cantilever and nude elliptical stylus.
And that last point, the elliptical stylus, has turned out to be critical for me when it comes to reducing distortion because I'm absolutely mediocre at turntable alignment, and the elliptical and conical stylus shapes are more forgiving of a little slop, as compared to the advanced line contact types.
Which brings up my last philosophical point:
A well-aligned and properly loaded modest cartridge will beat a poorly aligned, improperly loaded expensive cartridge.
I've heard some really awful, really distorted, not-at-all-flat turntable/arm/cart combos that were also shockingly expensive, yet were set up by bozos who tried to do it all by ear.