Humans have been steadily eating most other animals to death

Temperature records are inferred from climate proxies. Ice cores are the primary proxy used. Scientists manually count the number of air bubbles in a sample of ice to estimate the level of CO2. The amount of oxygen isotopes present is used to estimate temperature. The accuracy of these methods cannot be validated using current ice cores because of the time it takes for the layers to form. In other words, data used in climate science is extrapolated and estimated. As you can imagine, this introduces a great degree of uncertainty to the data. Nevertheless, this data is used as input to climate models.

Climate models are pieces of software written in Fortran that contain 500,000 lines of code. Models are run backwards to measure how closely they match the "observation" above. Stock market algorithms use a similar method of backtesting data. The problem with stock market models is they are only accurate with the historic data they are based on. Future price movement can simply not be predicted by using a few of the millions of variables that exist in the market.

Do I understand correctly; that humans ate first the bigger animals to extinction and moved to next smaller ? Isn't this counterintuitive? Bigger animals need better traps and (killing) tools while smaller ones can be catched by hand and can be choked ?

If I had to choose between a moose and a rabbit to hunt, I would go for the cute little bunny.

But don't take my choice as guidance, I grew up and lived in city.

Seems likely that destruction of habitat is to blame not eating them.

Same difference I suppose