Since we're specifically talking about SpaceX and Starlink, let's talk about Falcon and StarShip rockets. Let's look at the "long list of very nasty things".
Falcon 9 is expecting to launch a record number of flights this year, recent reports suggesting 52. Falcon 9 burns RP1 (a refined kerosene, very similar to jet fuel), a hydrocarbon that mostly produces CO2 when burnt, but also Carbon monoxide, and plain old carbon. The Falcon 9 has about 200k litres of fuel when fully loaded. It sounds like a lot, but for perspective a fully loaded Airbus A380 has a fuel capacity of 380k litres. So, they output of a record breaking year will be the equivalent of around 25 long distance flights of a large passenger aircraft. I'm not going to find out how many long distance large plan flights are happening, but many estimates are over 100k a day. So a record breaking year outputs a fraction of a percentage that airline flights do each hour. And the "nasty" stuff is mostly CO2, CO, and C.
Starship will burn Methane, and will burn even cleaner than Falcon 9. It will also be fully reusable, resulting in less waste again. On the downside, methane is a potent greenhouse gas, so any unburt fuel would be not great. That said, rockets are designed to make the most efficient of fuel, and any unburt fuel is not going to be released in normal operation.
> I'm no rocket engineer
Colour me shocked!
As for going through "every layer" of the atmosphere, I wouldn't get too worried about that. A lot of a rocket's fuel is burnt at lower altitudes (the same level at which passenger and military Jets operate) and the various zone mix readily, and are really so massive that you would have trouble measuring the impact of hundreds to thousands of flights of Starship per year.
https://space.stackexchange.com/questions/13082/calculate-fa...
https://www.sciencefocus.com/space/what-is-the-environmental... https://www.sciencefocus.com/space/what-is-the-environmental...
https://space.stackexchange.com/questions/32002/what-percent...