Interesting topic (the question was raised several times by my students as well).
My short answer is that adding the lower bound through a cut seems a good idea at first glance, but it creates a very large “unnatural” face where your search is trapped for a long while. Essentially you lose the objective function grip, and do not gain anything.
Let me explain. Take a MILP, and let LB be the lower bound at hand.
ASSUMPTION: LB is not tight at the integer optimum (otherwise, as already noticed, you have essentially a “feasibility problem” hence LB can indeed be very useful to stop as soon as a heuristic found a solution with cost equal to LB).
I ask you: what could be a positive effect of adding the cut
$$(1)\quad c x \ge LB$$
to the original MILP model?
Take a generic node of the enumeration tree, where some cuts have been possibly added and some variables have been fixed. Solve the corresponding LP relaxation without (1) and consider its optimal value $z$ (say).
If $z > LB$, then (1) is slack and hence useless. If $z \le LB$ the node lower bound would apparently improve (i.e. increase) going from $z$ (without (1)) to LB (using (1)) but this is again useless as you will never prune this node because of lower bound due to the ASSUMPTION above.
In other words, (1) will never help pruning a node, which would be the main reason to use it.
Instead, the negative effects of using (1) include (as already discussed):
A) huge dual degeneracy $\rightarrow$ you zig-zag between tons of LP solutions with the same cost (=LB)
B) blindness wrt the objective function at the root node and for many many other nodes $\rightarrow$ you are wasting 50 years of clever ideas such as pseudo costs, best-bound search, etc.
All in all, I would not expect any improvement when adding cut (1) to the MILP formulation—barring performance variability of course.