The 600EX-RT may have RF wireless control and triggering, but the 5D3 does not have RF triggering built-in. You would need to spend an additional ~$265 on a Canon ST-E3-RT to use the 600EX-RT wirelessly off-camera, in addition to the ~$550 for the 600EX-RT itself.
You can get used 580EX II's in great shape for ~$300 used, then add a ~$70 set of Yongnou YN-622C RF triggers if you want wireless off-camera flash. But there are so many good third-party options it is hard to recommend the 580EX II or the 600EX-RT unless you're philosophically opposed to considering anything but Canon gear.
For example, the Yongnuo YN600-EX-RT is functionally equivalent to the Canon 600EX-RT but costs only $125, while the Yongnuo YN-E3-RT is equivalent to the Canon ST-E3-RT transmitter but costs only $90. You could buy four YN600-E3-RT flashes plus a YN-E3-RT transmitter for LESS than the cost of one Canon 600EX-RT.
The Cheetahstand/Godox/Flashpoint AD360 is a high-powered, bare bulb flash powerful enough to overpower the Sun when shooting outdoors and costs around the same as a Canon 600EX-RT. However, this is a manual-only flash so you need to set the power settings and camera exposure values manually using a flash meter, or calculate/estimate and test to get the proper exposure.
The Cheetahstand/Godox/Flashpoint V860C is a traditional form factor flash that provides full ETTL (automatic exposure control) but uses a LiON rechargeable battery pack offering quick recycle and longer battery life than AA primary or rechargeable batteries, and also can be integrated with RF triggers compatible with those used by the AD360. So if you think you might want to add one or more AD360's in the future but want ETTL as well, this is a good choice at around $180 each.
Finally, if you are okay with manually-controlled flashes only you can't beat the value of the Yongnuo YN560-III or -IV with the YN-560-TX transmitter. The flashes cost around $70 each and the transmitter only $45, so you can put together a 4-light setup for around $325.