The change in comclemature from assault guns to tank destroyers showed the changing fortunes of the Germans as the war progressed. Ditto for the Americans. Assault guns are primarily used on the offensive, tank destroyers on the defensive. Tank destroyers tend, with their usually fixed casements, easier and cheaper to produce than tanks with their rotating turrets.
The E-10 might have been a little on the lightweight to carry either a 17 Pdr or a 77mm. The Archer was 15 tons, the E-10 was designed to be 10-15 tons so it might have just been able to absorb the recoil. It would have been rather cramped and whereas the crew could get out before the gun fired on the Archer, they may have found it a bit difficult on the enclosed E-10.
The various British recoilless guns were designed to replace conventional towed AT guns in infantry battalions. They were initially designed to be towed (BAT) and as it was realised there was no need for heavy gun shields and mobility became more important, they ended up being designed to initially portee'ed (MOBAT) and then fired from their carrying vehicle (WOMBAT). The problem with redesigning the E-10 (which was intended to replace the Hetzer) to utilise a recoilless weapon is that you either have to make it open-topped (with the recoilless rifle mounted outside the vehicle) or design a specialised turret (with autoloader).
As for it being used by 3 RAR in Korea - a recoilless version is unlikely. The BAT was only just adopted by the British Army in 1953, as Korea ended. For most of the war, 3 RAR were on the defensive, after the initial "rush to the Yalu" and the Chinese counter-offensive and the 17 Pdrs were more than adequate for that (although, I've never found much reference to their use, except as general artillery in the official history). They were mostly dug in, on precipitous ridges and the 17 Pdr couldn't be hauled up there, so I doubt an SP could reach them (even Centurions which were part mountain goat found it difficult to climb to the top of those ridges).
If you proposed (as has been suggested) that some German designers came to Australia post-war and proposed the building of the E-10 here, I suspect you'd also need for a recoilless version, some of the German recoilless gun designers to come as well. If you wanted it to be adapted to carry a 17 Pdr, I suspect it would need some enlarging to accept the bigger gun and it's recoil. Neither are beyond the realm of possibility IMO. I doubt though, it would have been in time for Korea, except perhaps at the very end.