What you've asked for with the superlatives in all categories is not possible for your budget. You'd need a pro-style hood with a blower capable of handling the effluent from your particular range. The general rule is to have the vent hood at least 6" wider than the range top. This is a great thread you can read to start your research: https://www.houzz.com/discussions/5161173/hood-faq#n=66

thank you Silverlined, that was helpful. At least I know I will not get everything I want. I also take it from reading this that I probably do not want one where the baffles are completely flush with the bottom of the hood. Perhaps one where they are tucked up into it would work better?

@mrmears That is precisely right. The hood should have an adequate under-hood containment area to handle effluent that outpaces the blower. It is not as much an issue for say, boiling eggs, but it is what makes a hood effective if you were searing steak. The most important feature is to have the hood ducted to the outside. Ducted hoods will usually extract more effectively than recirculating hoods. There are a few exceptions like in Europe where they have exchange air systems located next to advanced recirculating hoods, so the air still gets exchanged to the outside but in different ways than a ducted vent hood which is more common in North America. Wolf has a great reputation for effective hood design. I found an old display model at a showroom that I bought at a good discount.

I am having a hard time finding one with any under hood containment area! They all look completely flat.

If a Wolf vent hood is out of reach but you could still increase your budget, this one from GE Monogram is a compromise. It isn't a full pro-style hood, but it does have a containment area: https://appliances.monogram.com/us/specs/ZV750SY 

I like vent-a-hood for good extraction I do not think you need a bigger hood than the range if you get enough CFM for capture there is IMO no quiet hood fan but the fact is you turn it on when beginning to cook and 5 minutes after you finish turn it off so live with it. That GE monogram might as well be a MW for all it captures .

Our hood is so quiet we forget to turn it off sometimes, so there is such a thing. We have a Fantech inline blower & duct silencer. Works like a charm. Highly recommend it to you.

Capture and containment are two completely separate functions, explained in the thread both I and the OP referenced above. Note that I did not recommend the GE as first choice. A Wolf or other pro-style hood will have both larger capture area and deeper containment cavity than most residential hoods that are flat underneath. The model above is a compromise but would still exceed the OP's budget. I did also recommend looking for showroom display models on clearance at a discount to achieve the OP's goals with this purchase.

has anyone on this board every had a zline? something like this maybe https://www.zlinekitchen.com/professional-stainless-9597

The shape on that one looks good! Be sure to check out any noise reviews on the ZLINE blowers.

ZLINE says they are very quiet, but I would expect more noise than promised due to cost.