Saying that Aluminum is more conductive by weight is not really helpful, because it is so light. By volume, which is how wire is chosen, aluminum is a worse conductor. A larger size must be used - generally two AWG less (smaller number is bigger) than the proper copper wire.
For home installation you are right, the weight is not that much a factor, that is a big factor in why it was abandoned (instead of fixing the reliability problems).
But if you are talking about that needs to move (some car starters used aluminum stator winding, as that is made of very thick ribbon anyway so the mechanical robustness is good, it saved some mass the car then has to carry) or it needs to be supported (wires on power poles are for exactly that reason usually made as a steel core to provide the tension stress with aluminum wires around it as the main conductor), it becomes an important factor.
Same for some HID ballasts - lower mass on top of a tall pole quite simplifies things with the installation. Yes, the reliability problems made it not worth the weight savings, but the motivation was there...
Then who cares the diameter is larger, when the mass is what makes the handling of large size conductors difficult (what makes the difference whether you can handle it by hands e.g. during installations or need lifting equipment for any manipulation, or how expensive and complicated that equipment would be).
