Cannot wrap my head around setting up my Canon 430EX III-RT speedlite with this Yongnuo YN622C-TX trigger: whatever I try, the flash LINK light remains red, i.e. it is not connected to the trigger. Both devices are set to the same channel, the flash is set as radio slave, but it just does not work. Does anyone have this setup working?
The 430EX III-RT's built-in transceiver cannot link to the YN-622C-TX. The Yongnuo 622 system is a separate radio system that's incompatible with Canon's RT radio system. While both operate on the 2.4 GHz band, the signal protocols are different. It's kind of like how wi-fi, bluetooth, your garage door opener, and baby monitor all use 2.4 GHz, but don't communicate with each other.
To use the 430EX III-RT together with a YN-622C-TX, you'd need to add a YN-622C transceiver to its foot. Or, if you want to use the built-in transceiver in your 430EX III-RT, you need to get a transmitter unit that "speaks" RT, like a Canon ST-E3-RT, Yongnuo YN-E3-RT, or a Westcott FJ-X2m (aka a Jinbei TR-Q7).
But. What I would recommend, if you're happy to ditch OEM and go for cheaper 3rd-party gear? Is returning both the flash and trigger and getting a Godox TT685-C (or possibly V860 II if you need extended battery life) and XPro-C or Flashpoint R2 Pro II-C. While Canon RT gear is great, and you'll definitely get the best service and compatibility with OEM flash gear, if you're getting into off-camera studio-style lighting, Canon's OEM and Yongnuo's 622 system is limited to only speedlights, and only Caonn/Nikon TTL/HSS support. If you want bigger lights, if you find the grass is greener over at Sony or Fuji, you're kind of screwed. Or you're going 3rd-party anyway.
Yongnuo's 622 system is good, too, and low-cost, but it's kind of a nightmare if you ever want to expand to bigger lights or add on cheap manual-only single-pin flashes for multiple light setups, since their YN-560-TX system is also separate and (mostly) incompatible with their 622 system.
Godox has a full lighting system, with lights of all sizes that scale from a mini speedlight all the way up to a 2400 Ws pack and heads and a ton of stuff in-between at all pricepoints. And it's all in the same radio triggering system. that supports TTL/HSS for Canon, Nikon, Sony, Fuji, Olympus/Panasonic, and Pentax; and that support works cross-brand. So sharing the same lights between two different camera systems is easy. The Westcott FJ system is similar, but only has one speedlight and two mini strobes in the system at the moment.
And. The Godox pricetags are similar to Yongnuo's (although, sadly, the lack of customer service is similar to Yongnuo's, too. Buy from a reputable retailer that will support the warranty period for you and offer customer support. In the US, Adorama and B&H do this.) Westcott has better customer service, but higher pricetags.