Does anyone know what this saying actually means in relation to oil and renewable resources?

"it wasn't running out of stones that led to the end of the stone age"

Well, people stopped using stone weapons/tools because better technology became available i.e. bronze.

They are relating to the fact that we shouldn't wait until we run out of oil to change to a better technology. However the analogy is not perfect. Bronze is better than stone in almost every way, but there are drawbacks to many of the alternatives to fossil fuels.

"it wasn't running out of stones that led to the end of the stone age"
In short this means that views, practices, habits, and/or periods can change irrespectively if they themselves are obsolete or not. There may be many outside reasons to change to other approaches.
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... what (does) this saying actually means in relation to oil and renewable resources?
That you do not have to (or should not) wait till you run out of "stones" (options or raw materials).
;)