The thinking involved in driving a manual is very minimal once you get used to it, so I reckon any safety issues caused by that would be outweighed by a reduction in the unfortunately common situation of unintended acceleration crashes. You are lot less likely to drive through the nearest wall (or kids) if your instinctual reaction to moving when you should not be is to also go for the clutch and cut power instead of just pressing harder on the wrong pedal.

I taught my kids to drive using an automatic transmission, then once they were good at handling traffic, and all the more difficult parts of driving I let whoever wanted to learn to drive my stick shift. But I personally feel safer, not less safe, when I have to pay attention to operating the car, and as things stand now, think it makes me a better driver - that may not matter once cars are more advanced but for now, with driver-operated cars, stick shift is the right level of engagement for me to not get distracted. I do hate shifting motorcycles though, because I am not as familiar with it and it makes me feel less safe. Which is how I imagine OP feels.

See this is so alien to me because I've been exclusively driving a manual my entire adult life and I don't think about it. No more than I think about which pedal is the gas, which is the brake, and which direction the car goes when I turn the wheel this way. I just drive.