Why fire? Because, as the GP said, it lets you use energy that isn't your own physical, biological energy. It lets you "level up" in a way that a muscle-powered tool doesn't.
What they didn't say: It also lets you smelt metals. That lets you build much better tools.
Until the industrial age, I think fire was less about having more energy to work with and more about things that could never be done with any amount of mechanical power: heat, light, cooking and all its many benefits, repelling nocturnal predators, and transforming available materials in many useful ways (including smelting but also e.g. fire-hardened wood).
It did also replaced mechanical energy in some ways, e.g. hollowing out bowls or canoes, primitive mining by using large fires to crack rock faces. But I feel like those are less transformative than the other effects.
Also fire will be a great help in figuring out other chemistry.
There's a hypothetical alternate path where you build the "head end" of civilization using advanced biotechnology, synthetic organism, fermentation, 3-d printing and such. It's an attractive path for human space colonists but how would you figure out the genetic code without metal, glass, computers and such?
I think that we're kind of biased about these sort of things due to the limitations that evolution has placed on most animals.
We take for granted that we cannot regrow lost limbs or direct individual cells to grow into arbitrary organs as directed by our brains because we can't do it but there's no intrinsic biological reason why that must be so.
In a hypothetical first contact situation with intelligent life as we know it they may find it absurd that we can't do those things and wonder how we reached the technological level that we have despite having to spend so many resources on hospitals, work place safety and mechanical R&D instead of just growing what we need what we need it.
Our understanding of things like metal, glass, and computers would be totally different and far more implicit if we could simply grow organs or organisms that produce things for us on a molecular level if evolution had granted our brains the ability to control individual cells of our body.
If a species were committed to very long survival or space travel there is a good chance, I think, they would consider making changes to their own biology.
For instance most of the places that are "habitable" in this universe are on water moons or outer solar system/interstellar bodies that have water inside because of pressure and tidal and geothermal heating. A motivated enough race could create some species (is this the right language?) that would represent itself to take advantage of these sorts of habitats.
Yeah I think that's where we're going too. Just like multicellular organisms are metaorganisms that bring all the advantages that this entails the next level will be metaspecies.
It's going to be really fascinating when we unlock the true power of single cells and an individual can use those cells to generate any sort of body type that they want or any sort of complex structure to build things out of.
Imagine structures built out of bone or enamel. It sounds kind of crazy but it isn't, We already build structures out of wood and we insulate them with wool or down.
What they didn't say: It also lets you smelt metals. That lets you build much better tools.