Indeed.
Aluminum is actually a (far) superior conductor to copper per unit mass. It would be used on transmission lines even if it was the same price as copper, because the towers can be cheaper and farther apart. It's in increasing use in EVs due to the lower mass.
Copper is still used when the conductive density matters, like the windings of an electric motor. But if copper prices increase further, manufacturers will make sacrifices to efficiency and power density in order to save cost. And they'll figure out how to better balance the use of Al vs. Cu, perhaps using Cu only for the conductors closest to the core.
We also use copper for transformers, which are fairy "dumb" in their usual design. Solid-state transformers exist, which use much less copper, but are currently more expensive. They will no longer be more expensive if the price of copper goes up too much. And they'll probably get cheaper in the long run anyway, regardless of copper price, in the same way that switch mode power supplies have totally replaced linear supplies in the consumer space.
I've seen increasing use of copper in fairly mundane uses, like computer heat sinks, that used to be aluminum. The performance is a little better, but it won't be worthwhile if copper gets way more expensive. They'll just go back to aluminum, or use some other innovation (carbon heat spreaders, etc.) if price becomes an issue.
We even use aluminium on "dumb" transformers for power transmission. Dry-type transformers tend to be physically larger because they use air and resin (rather than a tank of oil) to insulate, and so the major downside to aluminium conductors (needing a larger cross-section to carry the same current for the same loss) is no longer a limiting factor performance-wise. In most substations, and extra 20-30% physical size of the transformer is a fine trade-off for cheaper construction.
There has been substantial development in a replacement for all copper wires in the form of CCA - copper clad aluminum conductors.
This product combines the advantages of both materials - low price and mass, stable connection, can be soldered etc. while using a small fraction of copper. It's making inroads into aviation and motor vehicle harnesses, and will probably be the default low cost option for new homes in a few decades.
> I've seen increasing use of copper in fairly mundane uses, like computer heat sinks, that used to be aluminum.
Copper heatsinks go in and out of style... Copper heat pipes have stayed en vogue, but typically embedded in aluminum blocks.
I'd suppose the fashion goes somewhat with the price of copper, though I haven't tracked it. The heatsinks themselves have gotten far larger as CPUs and GPUs have gotten more power hungry, not to mention RAM and SSDs. A material that's a good tradeoff at one scale isn't necessarily one at a different scale.
At any rate, one should expect many of these trades to go the way of Al if Cu gets more expensive (which it might not). Not all of them, but we'll probably see a bias towards physically larger systems in cases where space isn't at a premium. And also a bias towards active systems over passive, liquid cooling over air, and so on.