You're confusing two uses of the term 'regulation'. Tap changers and voltage regulators actively sense the terminal voltage and adjust 'something' to maintain the voltage within some design limit. That's a 'regulator' and provides 'regulation' of the sensed voltage.
But 'regulation' also is a term used to describe the inherent voltage drop in some devices. For example, if you review DC generators, you'll find that simple shunt-wound generators have fairly good 'regulation' and their output voltage only drops a few percent from no-load to full-load when supplied with a fixed field. A cumulatively-compound DC generator (which has a series field and a shunt field), can have a nearly flat voltage curve from no-load to full-load with just a fixed shunt excitation, or even have a voltage rise depending on the degree of compounding. (of course, an active voltage regulator can counteract whatever inherent regulation a machine may have)
In the case of simple fixed-tap transformers, the term 'regulation' can be used to describe how much the output terminal voltage changes from no-load to full-load if the primary voltage is held constant. This use is less than perfect as it is much better to use the transformer's impedance along with the load's power factor to get a more precise answer.
In the US, voltage regulation is accomplished with load-tap-changers, capacitor banks, and other 'voltage support services'. But just like in Europe, it is done at the substation or higher level and not done at the typical distribution transformer. There are exceptions for rural areas though where the line length of the primary leads to some issues.
daestrom P.S. In the US, a 'tap-changer' may be built for either for unloaded or loaded operation. The 'unloaded' type can not be stepped to another tap while there is load on the unit (although it can still be energized). It's switch contacts cannot interrupt load though, so if you try to move it while loaded, you can burn up the tap-changer. The classic 'load-tap-changer' is actually several switches that are controlled in a precise sequence to shift the load from one tap of the transformer to another while not interrupting the load current.
P.P.S. Load tap changers typically have a significant time-delay built into the controls so they do not 'hunt' or respond to short drops in voltage such as starting a large load. 15 seconds to several minutes is typical. So even with load-tap-changers, starting a single load that is a high percentage of the system capacity will *still* result in a voltage dip.