
SEBO
AIRBELT E3 Premium
Ultra-durable, 10-year warranty, but expensive with short cord.

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Any Dyson handheld vac - in particular the Car/Boat vacuum (#885609040349). It's around $225USD. Definitely BIFL.
If you're willing to do the staircases in batches, I'd consider a dyson car+boat as a handheld-only option. I don't often recommend dysons but for stairs, this would be a decent choice. Should perform well and filter better than most of the other cordless dustbusters out there. When used on normal setting you should be able to get through a staircase at a time or maybe one and a half. A more expensive but better choice would be the miele duoflex. It's more solidly built and has the bonus of being a stick vac. If used with It's main brush head, you can cover the stairs more quickly and it'll do a really good job. The brush roller is really substantial. You could also get the total care model if you would prefer to have a smaller brush head accessory. I had another look and there really aren't any corded options I can emphatically recommend. Not for carpeted stairs anyway. There are some decent mini canister style machines which can be carried with a shoulder strap, but I find them cumbersome as well.
I had one of each, purchased long ago at the same time. The 3 motorized wands worked on car+boat sometimes and would not spin sometimes. Never had that issue with the unit they came with.
What a patootie!! Dyson hand held
Nothing wrong with using a shop vac as a regular vaccuum cleaner. it gets the job done. I think they are just designed to work in dirty messy shops, so they are bigger, and maybe don't have as good dust filters on them. When indoors, you want to avoid blowing dust and allergens all over the place by using a vaccuum with a hepatitis filter. You can use a regular vaccuum with hos attachments to suck up cobwebs and clean fan blades and stuff. Yes, we have two vaccuums. actually 3, but who's counting? A Dyson cordless handheld. and a Dyson canister vaccuum. We like having one on each level. I HATE dragging that thing up the stairs.
I have the Dyson hand held, the battery capacity drove me nuts. So I bought the 18v battery adapter, so now I use my Makita battery and can easily do the whole flat.

SEBO
AIRBELT E3 Premium
Ultra-durable, 10-year warranty, but expensive with short cord.

Numatic International
Henry Canister Series
Budget king, super durable, but heavy and poor on carpets.

SEBO
AIRBELT D4 Premium
Durable, powerful, long cord, but heavy and pricey.

SEBO
FELIX Premium
Durable, powerful, good for allergies, but heavy with awkward attachments.

Miele
Classic C1 Cat & Dog PowerLine
Durable, powerful, quiet, but very expensive, short cord.

Ranked #1
SEBO - AIRBELT E3 Premium

Ranked #1
SEBO - AIRBELT E3 Premium

Ranked #1
SEBO - AIRBELT E3 Premium

Ranked #1
SEBO - AIRBELT E3 Premium

Ranked #1
Numatic International - Henry Canister Series

Ranked #1
Dyson - V15 Detect Series