The usual approach for 2.4 GHz is to open the product, identify the radio chip, and use a USB logic analyzer on the SPI bus that usually connects it to the MCU to capture the configuration and data traffic, and figure out what the configuration register accesses mean from the radio chip data sheet.
The occasional appearance of combined MCU/radio parts prevents that approach, but often that is only used on one end of the link and the other will still have a distinct MCU and radio with an SPI bus in between.
This is heavily practiced in the radio control community, as people want to use their nice transmitters to fly cheap toys. And there is code out there to transmit compatibly with some of the odder chips like the X297, using more common ones like the NRF24L01 or its workalikes from Beken, etc.
A key advantage of SPI sniffing versus over-the-air sniffing is that the usually implemented channel-hopping doesn't mean you loose the signal ever time it moves to a new channel - and you actually get to the see the frequency registers being written. To capture a hopping scheme over the air, you would need something more like a spectrum analyzer to figure out the sequence, and then have software track the signal while you attempt to figure out the data. However, a remote which only sends occasional button presses may not use hopping in the way something sending data continuously (like an R/C set, or possibly a keyboard/mouse) would.