Help with picking new integrated for under 3k

If you can find a used Anthem STR integrated you would be set for many years, otherwise look at the moon 250i integrated amp.

sammyshaps, if you like the sound of your Rotel, why not their RA-1592 integrated?  It has lots of power (200w/ch) and that same neutral sound.  And its built-in DAC will put your Apple TV to shame.  The advantage of Rotel is that they have been around a long time and have a reputation for very high build quality.  I have a 15 year-old Rotel RA-1092 in a secondary system and it has never had an issue. 

If not the Rotel, I would concur with those who have proposed Hegel products, having recently auditioned their top of line H590.  Nice neutral sound.  Their H190 at 150w/ch @ 8 ohms (250w/ch @ 4 ohms) is no slouch.  But if you plan to upgrade other components you may want to think about how much power you will need for any future speakers you wish to purchase.  A 150-200w/ch integrated will drive most, but not all high quality speakers.   

I can't speak to the other brands suggested by others, but would encourage you to try to bring home a unit to audition in your room with your other equipment.  If you can find dealers nearby or others who will ship to you for demo, I highly suggest you do that. 

Happy listening! 

@ 3k the Parasound A21+ might be a sound choice (no pun intended) not integrated, but would be a nice foundation to build on for your large room, best of luck !

+1 on the Kinki EX-M1. 
You'll be amazed at the sound.

All the best,
Nonoise