I think you confusion arises from the fact that nowadays the steps of the analysis:
- solving with an FE solver
are all merged into one for a number of commercial design software solutons (Inventor, Solidworks etc). What you refer to as modelling, IMHO, falls into the domain of preprocessing.
For "real-life" problems, manually modelling the structure usually takes a lot of time. That is why a lot of companies have tried to unify the design software with the structural analysis, in order to automate the procedure.
In many cases, the automated solution work very well (quite often by sheer coincidence), that is why a lot of younger generations of engineers have developed a dependence on them.
Obviously, manually modelling the structure is quite daunting for an inexperienced user, however it gives the user with infinitely greater number of potential options.
The "modelling" gets intertwined with the FE solver, because you need to decide - as you said- about:
- type of elements (linear, shells, solid),
- the material type (...),
- contact definition
- loads application
TL;DR: You don't have to do the mesh generation. However, knowing how to properly do the modelling properly gives you much better chances of getting meaningful results.