You're reasoning in isolated abstractions: "biodiversity is good for humans"
That's pretty well accepted.
The argument you see in textbooks is that the interdepedence of ecological systems is such that at it is difficult to know what species are key to our own survival. so dropping biodiversity is like playing Russian Roulette. (Of course the rich will be able to supplement, but I mean key to our own cheap survival).
But more importantly to me is the intellectual resource. Each species comes with it's unique proteins and biological processes. Losing them without studying them is a permanent loss to our knowledge, and future study is more likely to have useful results than current study, as our understanding of the biochemistry allows fuller understanding and so utilization of the processes observed.
"tigers dying reduces biodiversity"
I linked to a paper with this (generalized) result: reduced genetic diversity withing species reduces biodiversity of other species in the ecosystem.
"ice sheets move slowly and allows migrations"
The species in current existence have survived the repeated glaciation cycle of the holocene. The current warming is more rapid, and in the wrong direction.
With that kind of superficial reasoning, you can "prove" anything in any complex problem by just picking out the right abstractions.
I think you're ignoring the proofs. If you think one of them is wrong, we can delve into it. But read the paper linking genetic diversity to biodiversity first.
Large predators are usually already evolutionary dead ends, candidates for natural extinction
That seems like an isolated abstraction. Do you have any science-based evidence of this claim?
Humans have killed off many apex predators in many environments over the past few millennia.
Can you give a few examples?
Generally, the main effect has been that human livestock and humans have become safer.
Do you have any science-based evidence of this claim? I think that it is wrong. When you remove an apex predator, biodiversity crashes.