Can I replace the 433MHz modules on my homemade remote controller with an NRF24 transreciever?

Building your own radio system is a good exercise for understanding and appreciating the technology, but always remember to include a failsafe.

Regarding swapping modules, it looks like you are using the 432MHz modules which use On/Off Keying (OOK.) The work essentially by sending the state of a digital line, and support some serial formats like UART but also arbitrary pulses such as Pulse Width Modulation (PWM) and Pulse Position Modulation (PPM.)

However, the nRF boards communicate locally using SPI which is incompatible with the above. Without knowing more about your design it is difficult to say for sure how difficult this would be to integrate, but you will need a microcontroller at both ends. I can’t make out what the existing chips are – if you currently have a MCU you may just need to tweak a few lines of code but if you don’t, a more fundamental redesign may be needed.