Are so-called "model independent" searches for new physics in particle physics collider actually "dependent" on the Standard Model model? Are so-called "model independent" searches in particle physics colliders meaning model independent of new physics ?
Are those "model independent" searches actually "dependent" on model, in the sense that the results are in the form of a limit on the cross-section of a given process, thus these searches assumes that the Standard Model is true in the sense that the Standard Model would allow this process to occur, thus being "dependent" on the Standard Model model.
 A: Let us say you have been born on a mountain. The first time you go a large distance from your birthplace you start mapping the new land, because your brain has a model on how the land would look from your living experience. A lake met on the way would not fit your experience of maps, the constituents of your brain map should increase, there are also lakes, not only waterfalls, wide rivers and finally the sea .... Each new observation expands the model in your brain on how the land is made up, with new discoveries.
That is the way physics has progressed, and  since Newton times, mathematics could be used to map the existing knowledge mathematically . Mathematics allows not only to map the data but also to accurately predict what new data should look like. If the prediction fails,  then a new, unexplained by the model, phenomenon has been found, which is what collider and other experiments are trying to find, what is              not-mapped/unexpected.

thus these searches assumes that the Standard Model is true in the sense that the Standard Model would allow this process to occur, thus being "dependent" on the Standard Model model.

The dependence to the standard model is of the background plotted in the measured distributions. Disagreements to the standard model curves indicate a new mathematical component should enter in the fit to explain the data, and the standard model should be extended or changed.
Due to the precociousness of theorists a large number of possible deviations and new theories exist, waiting to be tested. The standard model itself solidified to the present day version when the predicted Higgs particle was definitely found (could have had any mass within a large range of masses). The new phenomena  that are being searched are for predictions of theories beyond the standard model.
History of physics has shown up to now that when the standard theory of the time is tested at its limits, new phenomena appear and the extra knowledge   extends the technological possibilities of civilization.
A: My comment, as per request:
You are victimized by bad language. The "Standard Model" is an established Theory... the name is a mere historical throwback. "Models" are supposed to be conjectural speculative New Physics models.
So, yes, these are searches for new physics of any kind, at variance with the established Theory.
