In America electric rates are increasing due to 20 years of stagnation and failure to invest in transmission, and new data centers are on average associated with lower, not higher retail electric rates. This is because large consumers drive down marginal prices.
In your country things may be proceeding differently, but that's the story here.
This doesn't really hold when we're talking about discrete pieces of infrastructure that may need to be built. If the current transmission infrastructure is fine for the current users, and then a new large user comes along that necessitates new infrastructure, that is a huge new cost that simply that simply didn't exist without that large user.
I believe in many areas this is the kind of infrastructure investment that regulated utilities can easily pass onto existing rate payers, which is where the problem/narrative comes from in the first place. So your theory doesn't match the actual results people are complaining about.
Same thing when datacenters take power generation into their own hands and build new gas turbines rather than solar/batteries or paying to expedite grid construction. And I'm willing to believe these problems happen in a minority of cases, but the problem is that they seemingly do happen and existing residents are left without recourse.