I don't think we have any active craft capable of recovering it. The space shuttle probably could have done it, but with a cost of about $1.5b per launch, there is no way that would be worth it.
SpaceX's Dragon 2 easily has enough cargo capacity to bring it down (~3 tons vs 0.5 ton), but there's still the question of intercepting, capturing, and securing it in Dragon's cargo bay. That would still cost something north of $100m to recover the lander.
Dragon likely wouldn't be able to get it, unfortunately.
The lander would easily fit into the unpressurized cargo bay (the "trunk"), that is typically used to launch various vacuum-bound payloads alongside pressurized cargo inside the capsule. However, for a return from orbit, the trunk cannot be used - it is not protected by a heat shield, and is ejected before re-entry.
You are correct that the return payload mass of a Dragon would technically allow it. But you'd need to somehow get the captured object _inside_ the capsule, which may be possible via the EVA front hatch for something smaller, but not 1m in diameter like the Venera lander.
Starship should be able to do it, since it is fully protected by heat shielding and returns in one piece, not ejecting any modules in orbit. But that's quite a while from being operational yet.
No, the Space Shuttle could not have done it. It had nowhere near the ΔV that would have been necessary for an intercept and capture. The OMS had only a tiny fuel supply. We're talking about orders of magnitude difference here.
Most of the benefit of the Shuttle program, and manned spaceflight in general, has come from R and D on the launch process, or from the prestige and bragging rights of being able to launch humans into space. So once they're up there, you're getting your money's worth (or not) no matter what they do, you may as well do something cool like recover a historic satellite.
SpaceX's Dragon 2 easily has enough cargo capacity to bring it down (~3 tons vs 0.5 ton), but there's still the question of intercepting, capturing, and securing it in Dragon's cargo bay. That would still cost something north of $100m to recover the lander.