What you're missing is that Yongnuo has three different flash triggering systems:

1. YN-560-TX and RF-60x manual-only triggering system.
2. The YN-622/YN-622-TX TTL/HSS triggering system.
3. The Yongnuo clones of the Canon "RT" triggering system.

All three groups use different radio signalling communication protocols. Your flash and its built-in trigger is from Group 3; your transmitter is from Group 1. These two groups are completely incompatible which is why your flash won't fire unless you put another RF-603 II to act as a receiver on its foot. Flashes that are in Group 2 with built-in triggers (e.g., the YN-685) can be used with Group 1 triggers, if put into 603 mode, but they can then no longer be used as TTL/HSS slaves. You can also trigger a Group 2 622-TX transmitter from a 560-TX transmitter, but again, no TTL/HSS.

Group 1 triggers, like the RF-603 II, are "manual only". The only things they can tell the remote flash are to fire and (if you shoot Canon or Nikon) to wakeup. A YN-560-TX transmitter can also control the power and zoom by group on the flashes with a built-in radio trigger (YN-560III, YN-560IV, YN-660, YN-720), but not on a flash on an RF-60x acting as a receiver unit. So, you won't have TTL, HSS, remote power control, or access to other flash settings (like ID codes). An RF-605 can add group on/off capability, but that's it.

Your best bet is probably to get a YN-E3-RT transmitter, which does speak "RT" and can do TTL, HSS, groups, ID codes, etc. with an RT flash. Or, to give up on Yongnuo and get Godox gear, where they only use one triggering system for everything. So if you just want to add a $65 manual-only speedlight to your fancy TTL/HSS one, you can. With Yongnuo, not so much.

If your flash/trigger are in the same group, then the basic troubleshooting routine to look at would be:

1. Are all the batteries good/fully charged in everything?
2. Are all contacts/hotshoes clear and clean of obstructions?
3. Is everything set to the same channel?
4. Is the transmitter fully seated forward in the camera hotshoe?
5. If using an add-on trigger, is the flash fully seated forward onto the hotshoe of the trigger?
6. If using a built-in trigger, is the flash in radio slave mode vs. an optical slave mode or on-camera mode?

See also:

• What are the Yongnuo flash naming conventions?
• Flash Havoc's 600EX-RT description