Because it will change the mixture. Depending on circumstances eg throttle & mixture settings, Density Ht, Carby Ht/presence of intake icing etc it can be enough to give you an engine failure.
As for checking it immediately after take-off, I think we have a difference in terminology/procedures. What I give below are very generalised & often change depending on a/c type.
Vital or immediate actions: Things to be done as soon as an engine failure is recognised
*Carby Ht: ON
*Mixture: RICH
Fuel pump: ON
Fuel: ON or change tanks
Then:
*Pick a place to land & establish a flight path to get to it
*Get a MAYDAY broadcast
*Look after the pax
Troubleshoot checks:
In a nutshell try everything else that might get the engine running.
*Recheck the vital actions (in the initial...erm... 'excitement' it's easy to mis-select something.
*Try each mag. in turn
*Primer: Locked
*Exercise the throttle
*Exercise the mixture
So, in the EFATO primer isn't appropriate BUT in other cases where there is time to troubleshoot then it is.
Hope that clarifies things a bit.