A strainer removes particles which are larger than the strainers’ mesh/hole size. If you are pushing the strained material through then you are basically pulverizing the material. I see these as two different processes - not as though " to push through is to go beyond straining".
Using a strainer as a strainer will result in a clearer liquid - especially in the case of stocks. Using progressively smaller sized mesh strainers will result in a very smooth, silky consistency of cooked vegetables, pulses, etc. Maybe these details matter, maybe not - depends on the cook and the end result desired.
Pushing material through a strainer can create damage. If the mesh is quite fine it might rip. Mesh material may distort or pull out of its attachments. The finer the mesh the more difficult it can be to clean the residue of what was forced through.
If you want to pulverize the soft material AND remove the hard bits (seeds, bone bits) then a Foley or food mill works great. If there are no seeds or other very hard bits and the material is low starch then an immersion blender does a great job.
