There is red/brown dust where the steering arm attaches to the hub. Does this mean there is movement where there shouldn't be movement?

I have the same on one of my minis. I suspect movement also and will investigate when I have time.

_________________
1969 Cooper "S"
1967 Mini Deluxe
1973 Clubman Van (the fleet spare)
1978 ex 1275 LS ("Wizard" Eaton Supercharged)
2015 HSV GenF GTS(occasional drive & tow car)
2019 MINI F55 Cooper S

Suspect worn holes in the swivel hub or arm. If they are worn oval, tightening the bolts won't help, even with new dowels. I had to change a swivel hub once due to this.

_________________
DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

Are the dowels missing?

I was expecting something like that. Will be dismantling it soon.
At least all the bits are still available.
How tight a fit should the dowels be in the hub and in the arm?
If the arm needs replacing, should the other side's arm also be replaced too?

I doubt it. I'm about to strip it down.
It's a 1970 S I've owned since 1975 and other than replacing wheel bearings the hub is original. The L.H. CV is original but not serviceable now; the wheel bearings need replacing and the hub might be worn where the cups sit. Probably due for a refresh.
Oddly, the R.H. parts all good to go again.

Dowels are a neat fit, size for size.
Loosen the bolts slightly on the other arm, if you can move it sideways it has a problem too.

_________________
DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

Maybe the dowels rusted out!

The hub and steering arm are now dismantled. There is a rust issue.
Soaking the parts in phosphoric acid to remove the rust and see what is happening.

Dowell looks like it has some wear

You need round dowels, round holes and a good fit.

_________________
DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

Was thinking about using bearing retainer on assembling the new dowels. Are there any obvious problems that would create here?

Have sourced a good used steering arm and the hub holes are still a tight fit. The rear bolt was under a lot less tension than the other 3 bolts when dismantling them.

You can try Loctite 609 or similar bearing retainer. I used it successfully to retain a Timken front hub bearing so it should work. If you have trouble dismantling it later use a hot air gun.

_________________
DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

Thanks for that, Dr M.