Bluetooth doesn't require a custom remote control. You can use a smartphone.
The aforementioned project uses a HC-05 Serial Bluetooth module. This module won't be used here, because the ESP32 already has Bluetooth on board.
It doesn't take more than loading the code from "ESP32 Bluetooth Classic with Arduino IDE – Getting Started" into the ESP32 and execute it. The Bluetooth RC Car Android app willingly connects to the ESP32. Serial data is coming in. We got this working! This went easier than I thought.
The problem with this solution is that there's no tactile feedback. You have to look at your fingers, so that you know where to put them and at the same time look at the car. Practice makes perfect, I guess.
To keep things simple on the software side, we'll use the ESP32. It already has bluetooth on board. To get a better range (hopefully), a module with external antenna will be used. The breadboard friendly module, featuring the ESP32-WROOM-32U is one from AliExpress (€4/pce).
Using the Bluetooth library seems to eat up more than half of the flash space.