Personally it upsets my stomach in a way that drinking carbonated water and taking a caffeine pill doesn't. And from my limited personal experience, it was contributing to constipation even when controlling for the caffeine (as in I don't think the dehydrating effects of caffeine were to blame). That's anecdotal and a quick google search on aspartame or diet soda brings up so much Dr. Oz level bullshit like you said that I don't care to sift through it. I'm not dumb enough to think my anecdote is proof of anything, but I'm also not dumb enough to keep drinking it when it makes me feel like shit.
One overlooked thing: you can build a tolerance to sweet foods. If you drink mountain dew all day, or a diet soda that's flavored just as sweet, then you won't enjoy the sweetness in a good piece of fruit as much as someone who hardly ever eats sweets. This is a psychological and enjoyment issue, not directly a health issue, but I think the practical use of this phenomenon for improving your diet is obvious: you can still enjoy eating sweet things while eating healthy if you adjust your thresholds for sweetness.
Drinking soda with every meal, diet or not, makes this adjustment impossible. And it also makes it so all the shit that has sugar added to it goes completely unnoticed by hams. Like Sweet Baby Ray's, which has 8g sugar per tablespoon (a tablespoon of actual granulated sugar is 12g), is just a normal bbq sauce to them. And when they do want dessert, their standard drink has upped their sweetness tolerance so much that it takes an insane amount to satisfy them. Suggest to a ham that they eat some grapes to satisfy their sweet tooth? Impossible.
Technically yeah you can say diet soda is like a seasoning that has no actual impact on diet or health, but the common implications of a person consuming it should be considered. Put some pepper on your burger? Who gives a shit. Drink a gallon of diet soda a day? Probably resulting in some fucked up shit.
People need to hear that they should be drinking water, not that drinking diet soda is technically not unhealthy.