Help Me Choose (new) Bookshelf Speakers, and Integrated Amplifier for $1300 and under!

Yes, the Elacs are the in thing these days. They are fine at their price points but I don't see them as class leaders.  Try the little JBL 530, quite a step up, very nice bi-radial horns.

I used a NuPrime IDA-8 integrated with the ELAC UB-5s, thought it was a great combination of smoothness, imaging, detail and dynamics. I did not, however, use the analog inputs, as you would be with your Music Hall front end.
Julie Mullins in her Absolute Sound review of the IDA-8 amp, extensively listened to a vinyl source through the analog inputs and was so impressed she bought the trial sample. 
Occasionally, the ELACs go on special for $399 a pair, and I've seen demo Nuprime IDA-8s at $899.
Also, if you can find a dealer who carries both brands, you should be able to get a system price close to your $1300 target.

The HSU HB1 are very nice and pretty efficient. For amps, if you can find a used Rega Mira 3 integrated, they are a very good value. 

A lot of good choices. The ELAC's are very good. My choice of a integrated would be the Rogue Audio Sphinx. Clean, simple, 100 watt/Ch. class D power section with tube pre. Has a good phono section and headphone amp section. Highly rated. Some good used prices out their for this integrated....
Cheers,
Tim

Might as well spend half that and save the other half by going with Emotiva. Specifically, their ta-100 integrated and airmotiv-b1 bookshelf speakers.

https://emotiva.com/products/amplifiers/pres-and-pros/ta-100

https://emotiva.com/products/speakers/airmotiv-b1