If you have 2 flights of stairs that's where I'd generally recommend the infant carrier TBH! But perhaps a lightweight one. Particularly for winter - you shouldn't put babies in car seats in bulky winter clothing, as it prevents the straps from being nice and snug to their body, so it can be practical to get them sorted and tucked in with blankets (over the straps) in the house and carry to the car.
I think the point about dropping the baby on the stairs is a bit far fetched, though. A car seat is designed to protect babies from car crashes, not falls. And I've never heard of anybody dropping their baby on a flight of stairs anyway.
If you did want to use the 360 seat from newborn I wouldn't recommend all of them, they aren't all the same and they don't all have very good newborn support. The seats I'd recommend from newborn would be:
Joie spin seats (any of them) - but be aware it can be really hard to get the harness secure for a tiny baby as they don't have much of a neck and the pads on these seats are huge and seem to overwhelm them. If you choose this come back and I'll explain how to do it!
Maxi Cosi Mica - I recommend getting the in between insert (Maxi Cosi Pearl insert). It's a big change from newborn insert to nothing.
Recaro Zero 1 or Salia Elite - both Elite versions come with an infant carrier portion that clicks in and out of the main seat. Very high in car so no good if you have a smaller opening on your car door.
Besafe izi Turn / izi Twist - extra recommended if you have very angled back seats.
Axkid One+ - doesn't spin but will take you up to 125cm (about 6/7 years) rear facing. Again OK for angled back seats.
The last three options are expensive, but perhaps not more so than buying an infant seat + base and then a second seat up to 4/6 years, depending on which you chose.
Another option is to look at "modular" systems i.e. one base which takes an infant seat up to 75/85cm (about 10-20 months) and then later a toddler seat up to 105cm (about 4 years).
The disadvantage with both 0-4 years seats and modular systems is if you have another baby within 4 years they can't both use the seat at the same time. However you could then buy a larger seat for the older child.
WRT head flop - you would want to go in and check the angle of your back seats in your car. If they are any steeper than about 5-10 degrees you have to remember that means in rear facing mode the whole seat is tilted forwards by that amount so the baby can end up almost vertical in some cars/seats, and that isn't good for them at all. This isn't only an issue for 0-4 years seats, it can also be an issue for infant carry type seats, especially those installed on a base. A couple of bases have the option to tilt to counteract this kind of angle. Of the 0-4 seats, the Besafe and Axkid both have features which allow them to counteract tilted rear seats.
If you did decide to look at some carry seats instead/as well these would be the lightweight ones I'd recommend:
Joie i-Snug
Nuna Pipa Next
Avionaut Ultralite (sold as Venicci Ultralite in UK)
Maxi Cosi Cabriofix (nb this is an older seat and doesn't compare to more modern ones for safety/features, even though much loved and recommended)
Maxi Cosi Coral - has a shell you leave in the car and internal part you carry around.