I received my CC debugger (a cheap knockoff bought on AliExpress) and tried right away to flash my CC2530 module: success!
Just for reference, I used this documentation on how to flash using TI SmartRFR software and for the firmware itself, I built one based on Z-Stack 3.0.2 following this tutorial I mentionned before (you can download the hex file from the Files section: CC2530ZNP-with-SBL (Z-Stack 3.0.2).hex).
To check if my flashing was successful or not, I used TI Z-Tool software (don't forget to check "Scan all Enabled com ports (startup and manual scan)" in Tools > Settings > Serial Devices > Auto Scan).
Also, while I was waiting for my CC Debugger to be delivered, I looked around on the Internet for possible reasons why my previous attempts at flashing the CC2530 module failed. I didn't find anything relevant so I decided to check carefully my components and found out the solder joint of D5 on my Wemos D1 mini was cracked... A quick rework and I successfully flashed my firmware using CC loader library (for those interested, I also uploaded the bin file in the Files section).