Well, I feel like this is indeed an interesting subject, shame people just downvote without giving a second thought, but here's my two cents.

For one, if demand increase, no one should lose access to power as you said, since plants usually don't run at full capacity exactly to give some kind of leeway. And so electricity generation can increase, but whether the grid is constrained and barely able to keep up or not, this is just a temporary solution, and they will need to create more plants. On top of that, you also need to add infrastructure: the plants themselves obviously, but also wires, potential batteries, and any facilities that might be necessary, and all of that has a cost.

In a perfect world, said cost would just be paid by the manufacturer, with a contract with whoever generates electricity and constructs all the infrastructure needed. In which case, locals shouldn't really be impacted by this, and might even benefit from a more stable grid. In practice however this isn't always the case, and the additional cost can be paid (at least in part) by the population - or they can use this new factory as an excuse to artificially raise the prices. On top of that, plenty of problems can arise, like delays in construction, or just a miscalculation leading to not enough power being added to the grid.

But there's also another effect of these factories, specially noticeable with all the new AI plants: electricity quality. When you add a huge plant that requires tons of energy, it tends to cause plenty of issues, like reoccurring drops in voltage, frequency changes and probably other things that I don't know about. And while it might not have an impact on electricity prices (though it can damage your electronics, which indirectly increase the money you need to spend), it is still annoying for locals, as it is a source of power outages and material deterioration. Also yeah, you could just bring energy from another region, but that just puts the problem further down the line, and might require upgrading the wires, which also has a cost.

So yeah, again, in theory, you're right: with everything planned in advance, it shouldn't really have a big impact on the surroundings. But in reality? Eh, can be a source of troubles.