The resolution itself does not drain the battery for reasons you said, rather what is being displayed does (or how it is produced). In the case of Simutrans with its software render then display area in pixels converts directly to processor load and so energy consumption.
But also when the computer idles, to be able to provide stutter free, say 4k, video, and simple 3d all must be able to handle greater peaks, which limits power saving through undervolting, increases leak currents due to larger dies, requires larger heatsink areas, greater max currents from batteries. When dealing with baseline power of a few Watt this matters.*
*My outdated notebook has a baseline power of 8 to 9 W, measured with powertop. My wive's new notebook must be considerably more parsimonious, perhaps in the few Watt range. If the manufacturers claims are correct, 5.6 W under load, for playing HD video (45 Wh/8h).