What's the best integrated amp in the $1500-$2000 range?

If you are looking for something with a good phono section, I would highly suggest Rega. 

Many of which has been suggested have already made the shortlist in my head:

Rogue Audio Sphinx v3 - I've yet to find a negative review, most reviews say that it well outperforms its price tag. I also fundamentally like that it's all analog.

Peachtree Nova 150/300 - I've read mostly good reviews with a handful who weren't in love with the products.  While my preference is analog (focus on my record collection), I will need at some point add a streaming component for my wife/guests - so this would remove that extra cost.  Is the 300 that much better than the 150?

Musical Fidelity 2si/3si - I just think they look great and have been following the reviews for years.

@reubent  - I wasn't expecting the Rega to be your favorite in the list, I'm now definitely going to audition it.

Amps that nobody mentioned, and I was wondering why?

audiolab 6000a
creek evolution 50a/100a
Hegel H90/Rost


Lastly about the speakers - the amp is for a new setup.  I was hoping to keep the speaker cost under $2000, so I'm flexible.  I want bookshelf speaker on stands that must suit my wife's design criteria.

This is why I posted this in analog.  I'm focused on a new analog setup (haven't yet selected the phono player too) and was looking for some advice on (for starters) on the integrated amp.

I have the creek evo100a you noted as having not been mentioned above. I paired it with Kef LS50s and love it. I have a friend with a peachtree nova300 and we both preferred the creek when swapping them out... however both amps are good enough that it’s a subjective preference rather than anything overwhelming and you might find the opposite to be true for you. I got it because I wanted a small form-factor, one-box, simple system for the master bedroom but it has since moved to my home office. One minor quibble when it was in the bedroom TV/music combo system is that the remote can’t turn the power on/off. This was an annoyance when watching TV before bed but wasn’t the reason for the change (wife wanted to downgrade to a sound-bar for space saving/aesthetic reasons) and it doesn’t bother me at all in a music only system in my office. I’ve spoken to a handful of other creek owners and all speak highly of the sound quality compared to other amps they’ve owned. I have the Ruby DAC module you can add but not the phono module. The DAC module adds digital inputs, a FM tuner, and Bluetooth but not built-in streaming...so while the sound quality is really good for the price, it’s not up-to-the-moment connectivity for a DAC. Without the DAC module there are only analog inputs. 
Compared to the peachtree, I’d describe the creek as more smooth & musical but slightly less analytical/precise. Definitely a fun and non-fatiguing sound that I can listen to all day. Some worry about electronic gremlins with British electronics, but I’ve never had any issues with mine in ~4yrs of ownership.
Sadly for you, the price of the evo100a is outside the range you gave, especially once you add the DAC & phono modules if you intended for that connectivity to be included at your price point. I can’t speak to the 50a because even though they look almost identical, they actually use totally different amp technology with the 50a using class a/b and the 100a using class G amps. If you can find the creek evo100a in your price range and it has the connectivity you need, it’s definitely a very good sounding integrated. I’d guess the 50a would be too, but I really don’t know.
If all analog connectivity is an option and you can find a used one in your price range, I’d also recommend the Belles Aria integrated...no frills, but it sounds fantastic.

@msafrick - The Rega Elicit-R really surprised me. I read a review or two and it sounded like it would be a good match for me. Musical, with a really solid bottom end. I don’t want to use a sub, so I wanted an amp that could help my speakers produce good bass. That, plus having a good built-in phono stage, in a relatively small form factor, sold me. I bought mine used at somewhere around $1500 (+/- $200, don’t really remember). Anyway, once I got it I was really impressed by the build quality. It feels like it’s milled from a solid block of aluminium. One caveat, it does sound better after it’s been powered up and gets warm. Because of this, I leave mine on 24X7.

Good luck. I hope you find something that makes you happy!

You can easily get an awesome tube integrated amp in that $1.5-2K price from PrimaLuna or their sister company Mystere. I’m extremely satisfied with my Mystere IA11 amp from Upscale Audio in a vinyl system which is listed in "virtual systems". If you don’t want a full tube amp a contender for a hybrid w/ tube preamp and SS class A/B power amp (not class D like the Sphinx) is the Vincent SV500 integrated amp, also from Upscale or Audio Advisor.