AV Processor has no amps so you'll need to buy a separate amplifier(s). These are separates and you stated that you did not want to go that route. An AVR (AV Receiver) is an all-in-one solution so it will have amps built in.
Not familiar with the Anthem AVR lineup, but the MRX 740 should be sufficient for your needs if you want to go with Anthem. If you want to experience Atmos, you'll need to purchase some height channels.
The MRX 1140 gives you more height channel (6 in 1140 vs. 2 in 740) and is more in line with the cost you listed ($4200 for 1140 vs. $3100 for 740).
Anthem appears to have recently "updated" their AVM / MRX lineup to include 8K processing so make sure read the specs to be sure you're getting an 8K unit and not the "older" 4K unit.