Algorithms – Achieving Local Optimality in Simulated Annealing

I am currently reading about local search techniques. I understand that local search algorithms tend to be in local optima and therefore usually do not find globally optimal solutions. Therefore, there are more sophisticated approaches that allow us to maintain local optima and improve overall solution quality (eg, simulated annealing or genetic algorithms).

What I'm thinking of is this: As far as I understand, the simplest approaches, such as mountaineering (best-fit), will at least be guaranteed to find locally optimal solutions to the given neighborhood. Do not we lose this property when we use simulated annealing or genetic algorithms? Even for the first version of mountaineering, it is no longer guaranteed to find a local optimum and at the same time to have the advantage of a possible runtime shortening.

Is it a compromise between an increased ability to achieve globally optimal solutions (or a reduced runtime in the first-fit case of mountaineering) and a higher risk of not even getting a locally optimal solution?