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

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Okay, Kenmore makes a pretty decent upright, the Intuition. Model # is BU4022 or the slightly more expensive version #BU4050. They're the same vacuum in a different package deal. It is a bagged vacuum which also has HEPA filtration, the best at cleaning the air. The power roller can be switched off. You might have to look around to see where they are available in your area, but I've seen them on sale for $220. The one down side is the vacuum is 11 inches wide vs your Bissell at 13 insches wide, so a slightly narrower cleaning path. On the plus side, it'll be a little quieter than the Bissell. Good luck!
The Kenmore intuition is a great option, I just picked one up for family. Great suction, the rotating brush can be turned off, and it is a sealed hepa design. On sale I found it for 220. It's a bagged version so no mussing with a dusty bin. Only downside is it's 11 inches wide instead of 13 for large spaces, but should be fine in a basement suite.
Miele is not awful on bare floors. I've had 2 of them and currently own the C3 canister. You just need the right attachment for bare floors, a parquet. Kenmore is a good design but budget build, hence the higher failure rate. But they might not have the best tool for dusting, and the extension hose is a bit shorter. For dusting with the Miele, I'd suggest an after market dusting tool. So you could buy an "electro" Miele, which means it has an electric powered rotating brush for carpets. But it might not come with a decent parquet so buy this Miele part number: SBB 300-3. Then also buy this dusting tool, I have it and it works well: SUB 20. Not exactly the cheapest solution, but any Miele canister will last at least 15 years if you change out the bags and filters regularly. Or buy the Kenmore and it'll take longer for dusting, maybe you could buy the Miele dusting brush and and adapter ring to get it to fit. The Kenmore should last 5 years or more unless its defective.
I noticed that you now mention just having a couple of area rugs, I may have misunderstood your original post. I don't think you need an electro Miele at all, I would suggest something with a parquet tool and Miele's turbo head for your area rugs(Canadian Tire has some Mieles). Perhaps there is a Miele model in Canada that has both included, I'm not sure. That Miele dusting tool is a good one. The Kenmore would get you by if you're on a tight budget, with the proviso about dusting your electronics. I bought the Kenmore Intuition for my MIL, and cleaned out her basement with it. Really decent suction and performance for the money.
My mistake, this model does not actually have height adjustment. Kenmore's bagless vacuum, the DU1275 does have height adjust. No idea why they selected that model for the height adjust and not the Intuition.
I bought the Intuition BU4022. Are there any differences to your knowledge other than a pet attachment? Wondering if I should return my model for this one.
Kenmore bu40xx models get recommended a lot. Pick the variant that has the tools etc (color!) you need like 4050, 4022.
I get that it can be a pain to carry around a canister with powerhead. Definitely best to vacuum moving forward as you go. But stairs. Have to learn the least friction pattern such as start at the bottom, vacuum 2/3 of the flight, move to the top and vac the last 1/3. Then continue on forward. Another less gangly option we used to suggest to people was to buy 2 vacuums. Solid upright for carpeted areas + solid straight suction canister (no powerhead) for bare floors and above the floor. Straight suction canisters are much lighter and easier to move around. Almost any one will do a similar job. Something like a Mighty Mite or the purple Zing are super light and easy to store and pull out for quick clean ups. A simple upright is also easy to store and pull out without all the bells and whistles, hoses, tools hanging off it. You could also consider an upright like Kenmore Intuition, Sebo Dart or Felix any of which can do all surfaces. It can be a bit problematic to use the hose with tools on an upright as they have a tendency to tip over. And unless they have a dedicated bare floor brush vs just turning off the brushroll, I don't think they work as well as a canister. I do think a canister with powerhead is the best single machine multi-surface cleaner overall. Above suggestions are all bagged and corded.
How about a Kenmore Intuition BU4022 Bagged Upright vacuum cleaner? Corded, bagged, HEPA filtration, relatively small and light weight (14 lbs), good for carpets and hard floors.
Not sure on the Canadian market pricing but I read through a ton of posts here last week and bought a Kenmore BU4022 (same as the BU4050 but without the hair eliminator brushroll) and it is an incredible vacuum for $150. I was upgrading from a tired old Dyson DC14 and a Dyson V8 stick vacuum for hardwood, rugs, and some carpet and the Kenmore has better suction and cleaning power and a cleaner experience with the bag (no dust plume when emptying the canister). FWIW I chose the standard brushroll because a replacement is available should I need it in the future. I looked all over online and could not find a replacement part for the BU4050 despite it constantly being recommended here. I could be wrong but I read that the blades that prevent hair from wrapping can also lead to faster brush wear, so I chose the standard brushroll in hopes that it will give me better longevity.
Kenmore intuition It's a solid machine that is just a good vacuum overall, it should make you happy for a good amount of time! Check around, you can usually find it right at your budget. Even then it's worth going over by the $10 or so. There are a couple different colors with some other tools and stuff, generally the same machine however. https://www.amazon.com/Kenmore-Intuition-Friendly-Handi-Mate-Combination/dp/B08NPBCG97 Also, always do a good search at your local thrift stores and marketplaces for good machines that people don't want! To date I have found a Miele canister, a Kenmore Intuition, and then an older Progressive that I actually used to have years ago. *I know you don't want bags but the name brand Kenmore bags are like diapers, filter well, and last a long time. For your price range you will be hard pressed to find a bagless machine that will make you as happy as this one. Trust me, I've tried lol
If you want to save some money, get a Kenmore Intuition. They are quite good, have good filtration, and good performance all the way around. That is if you want an upright model. They do also have good bagged canister machines too. Miele is great but even the classic C1 Pure Suction is around $400 with the moderately better Guard M1 model being $500. Both of those are suction only so if you were to move or something and have carpets in the future, you can't get the power heads on them. (check your local authorized Miele vacuum stores, they often can help with pricing)

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.

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SEBO - AIRBELT E3 Premium

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SEBO - AIRBELT E3 Premium

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SEBO - AIRBELT E3 Premium

Ranked #1
SEBO - AIRBELT E3 Premium

Ranked #1
Numatic International - Henry Canister Series

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Dyson - V15 Detect Series