I shouldn't be saying this online, but I'm not lactose intolerant and I very occasionally drink black coffee which does give me the runs.

That being said, I only take it if I have something really important that I wanna be fully awake for, but only if its within the next 1-2 hours because I don't wanna be 3 hours into something feeling like a pressurized gas bag lol.

Eating food before or alongside coffee can help mitigate this effect, but most people have it in the morning with breakfast which is usually a light meal so it doesn't stop all of it.

Also the anxiety probably comes from the increased heart rate alongside a mental stress factor. If you're not worried about something, you'll feel awake but a bit lax. But if you're already nervous or anxious, coffee will multiply it.

Cleveland clinic says says about coffee's laxative effect:

Researchers found that 29% of coffee drinkers report a desire to poop after drinking coffee. The feeling can come pretty quickly, too. (In as little as four minutes!)

And about lactose intolerance (same article):

An estimated 65% of people have some difficulty digesting lactose, which can lead to restroom runs. Lactose intolerance can cause diarrhea and other gastrointestinal (GI) issues within 30 minutes of consumption.

It makes me poop, and I never use milk products with it.

Coffee is a diuretic so yeah if you over imbibe youre going to poop even if it's the purest black.

That isn't to discount your point that the stress lactose puts on guts isn't helping.