It would kill at _least_ hundreds of thousands and negatively affect the lives of an order of magnitude more.

It would almost certainly fail by any reasonable metric. Look at how occupying Iraq and Afganistan went, or how the US has done in South America over the last century.

And that's just for starters.

2) Could the US significantly damage the cartel(s) on their own turf?

3) What are the cartels? Are they the literal president-level politicians who are affiliated? Should the US "take them out" too?

I could go on but really the short answer is that the US would end up causing a mess, ruining the lives of normal people, and damaging its relationship with a major world economy and major trade ally.

It's an impossible situation, but I don't think it's difficult to understand why Mexicans would not want the US military to invade based on its track record.