Remove the old metal box altogether
The “fan light assembly” includes that ring which left in the ceiling, and also the metal junction box which the wires now (I guess?) come into.
Normally when you remove a can light, you remove that entire kit-n-kaboodle as an assembly. You carefully disconnect and pull the Romex/NM cables (unsheathed wires, bare wire ends and all) out of the metal junction box that is part of the can light. The entire can light assembly goes away and you are left with 1-3 NM cables and a hole in the wall.
Next, you re-route the NM cables directly to the new fan box. Then hook them up as appropriate.
If there are 2+ NM cables in the old box, one is wise to carefully mark each sheath in some way, and capture the old hookup method with markings. Either just put numbers on them and take photos, or I would write down their function - For instance the cable whose white wire is nutted to other black(s) is a switch loop.
Or, use the old junction box as a splice box—- oh, wait.
OK, here’s the disclaimer: You cannot “bury” a junction box in a way which is inaccessible. Now, can lights can sorta get away with that, because there is often access through the can opening, and in fact, UL specifically approves this application according to UL White Book standards. But your clever “re-imagining” of the design, well, that voids the UL listing. So now we must review the “accessibility” issue anew.
If you have attic access, you can use the box. If the only access to this box is the UL-prescribed method “through the can hole”, then no: your fan bracket blocks that route. You cannot use the old junction box as a splice box, and you must go back to my top solution.
So let’s assume we cleared that hurdle...
You need to get ground from the legacy steel box to the new fan steel box. I see 3 ways to do that.
Use the metal sheath of the armored cable. This will require getting an appropriate fitting to allow it to enter a 1/2” knockout.
Run a separate ground wire, bolting onto each box somehow at each end. The wire can either enter the steel box through a cable clamp and join the existing grounds, or it can simply connect to the steel box itself. Either through a bolt-and-nut, or the gold-standard way is to drill a hole appropriate to be tapped for a #10-32 screw, and then either tap it with a tap, or just use a self-tapping 10-32 screw. You’re also allowed to use 8-32. The thread pitch must be -32 or finer; no sheet metal screws allowed.
Replace the armored cable with familiar NM... with appropriate cable clamps on both ends. I don’t need to tell you how to wire that!