I'm going to give some unwelcome advice and say that based on your criteria, you can't get there from here. You'll have to adjust your wants if you want an ebike.
What do I mean by that? You said
"... bike that feels like a bike instead of a scooter, when the assist is not being used, to feel like a proper bike."
And that is a really tough one. Ebikes are heavy by necessity. You have to add in a motor, and batteries are quite heavy. To keep the weight down, you have to go small, which limits usefulness. When you are done putting everything on a diet, you are looking at a 40 lb bike that can't do much because you prioritized weight, and gave up capability to get that prioity. And even though you shorted yourself on the electrical assist side, you still bottomed out at 40 lbs, which is about double the weight of a decent road bike.
I owned one of these kinds of bikes myself, which is exactly what it is you are asking for I think:
The most Stealthy bike on the market, with a hidden battery and discreet mid drive motor... this bike looks like an ordinary road bike but packs a big punch.
lunacycle.com
I added a custom wheelset to mine and did a few other things to spiff it up. Saddle. Stem. Bars.
View attachment 8361
Price is smack in your wheelhouse. But to get down to that 39 lb weight they had to put in what amounts to a tiny battery. And the motor is a mid drive, which is a good thing, but its still going to peak in the 400w range and with the 3-speed setup, despite its mid drive credentials its only good for mild hills. Range was truthfully described by the seller (for once) at around 25 miles... which is the very outside of its envelope.
I sold it on because it was a bike, but precious little ebike. It DID feel like riding one of my old road bikes which was neat, but the joy of that wore off when I realized how limited it was in its utility.
Another option in this same vein (35 lbs) is the unfortunately-named FLX Babymaker.
Our best selling stealth electric bike is back and assembled in the USA. Features include a single speed Gates belt drive, compact LCD display and sleek polished frame. Powered by an integrated battery and quiet rear hub motor, this bike is great for your daily commute or urban adventure. **The...
flx.bike
A little more (claimed) battery, a little less money, a whole lot of baloney on the range estimation in that advertisement and a small hub drive with no derailleur so its really a single-speed electro-fixie kinda thing.
My personal feeling is you are going to have to adjust your perception of what an ebike is. They're bigger. An ebike is a bicycle-shaped-object. It will never be a bicycle and it needs to be treated as an entirely different animal.
(p.s. if you do that, $2000 is near the bottom in terms of price point. You won't be choosing between excellent options, but rather most-livable given the budget limitation).