I know that MILP solvers are bad with scheduling problems. However, if we are allowed to keep unscheduled some tasks (i.e a solution with 0 scheduled tasks is a feasible solution but we add the objective (makespan) the maximiation of scheduled tasks), does things become easier?
That's a misconception - MILP solvers can be brilliant with scheduling tasks as long as the modeler knows their stuff, but that's true of all NP-Hard optimisation problems. The snag is that a certain level of custom modelling work is typically needed for real problems, but this is not a solver limitation per se.
If what you have in mind is to make the scheduling of some tasks optional in order to aid the solver, then many solvers support lazy constraints.
This is not trivial to model properly because you still need to ensure that the tasks are scheduled some of the time, but to answer your question directly no, it's not necessarily easier. The only way to check is to try out both formulations and see what works best for a particular problem.
In global optimisation (including MILP), constraints are our friends because they help us reduce the solution space. However, the problem is NP-Hard because (or the other way around depending on your preference) there is no one-size-fits-all way to change the problem to make it easier. For some problems constraining the system more tightly helps a lot, for others it makes things much worse.
From experience, what does work much better than MILP if you need to have fuzzy scheduling is to formulate an MINLP.
What follows is conjecture.
If you have a tightly constrained model, for which the solver struggles to find a feasible (or good feasible) schedule, then I suspect that allowing tasks to be skipped (at a penalty) may facilitate getting a feasible solution. Whether it will get you to an optimal solution sooner is anybody's guess.
If the MIP solver finds feasible schedules in reasonable time but struggles either finding an optimal schedule or proving optimality, I suspect that allowing tasks to be skipped will make things worse. It will expand the feasible region (so the search tree probably gets bigger), and I think it likely will loosen the LP relaxation bounds.
Overall, I doubt I would try it. If the issue is difficulty getting good schedules early (or any feasible schedule), I would try either scheduling heuristics or a constraint solver (specifically one that has global constraints tailored to scheduling problems). With heuristics, I would try to get a good schedule and then use it as a hot start for the MIP solver. With a constraint solver, I would first try to let the solver progress to optimality, and only use its solution to hot start the MIP solver if the constraint solver looked like it was going to struggle to reach optimality itself.