There's a great article on project crashing in PMI that addresses your concerns.
In practice the most common requirement for project re-planning
calculations concern time and cost. Clients often ask for projects to
be speeded up and need to know how much of an increase in speed is
possible as well as what it will cost.
The analysis and execution of this time change, and its attendant
impact on cost, is commonly known as crash analysis. In crash
analysis, a project manager offers re-planning advice based on the
functional relationship between time and cost. The objective is to
look at that relationship for the process concerned and to generate an
alternative cost and time scenarios. The client can see how much it
will cost to meet a range of different time options.
Knowing the critical path in the project schedule is key. This will
dictate the current finish date for the project. It will also identify
the activities that need to be reduced if the project is to be
completed sooner that what was first envisaged. By assessing the
associated costs of reducing the critical activities, it is possible
to select the most cost effective crash sequence.
From: Kelly, Éamonn V. (2009). "Crash with confidence". Available online at: https://www.pmi.org/learning/library/performing-crash-analysis-alternative-cost-schedule-6750