No $800 receivers will be able to drive 4 ohm speakers in a 5.1 or 7.1 system unless the system is in a small room and you don't listen loud (say <85 dB./m).
Very few receivers can do well with 4 ohm speakers regardless of price.
Mmmm... My Onkyo TX-SR805 is doing a very good job with all my 7 speakers that are all below 5ohms, and down to a minimum of 2.8ohm (center channel).
My two main fronts are 3.9ohms over most of their range (down to 24hz), and I run them full range too!
And my 805 cost me less than $700. It was as low as $449 at one point.
Also my Denon AVR-3805 (more $) did a very good job too. And some Yamaha models are adequate too, without going for their flagship. My 3805 cost me less than $800 at the time. I bet you can buy the Yamaha RX-V1 for less than $800. The Denon 4306 should be add for around that price too.
You can get quite a few receivers for $800 and less (older models & refurbs) that will drive 4ohm loads to satisfying levels. You just have to know which ones.
Also all Emotiva amps (some for as low as $529) will do just fine with 4ohm loads. The XPA-5 is $799, for example. The UPA-7 is $599. The XPA-3 is $529, when in sale.
Yep, there are some bargain out there, when you know what you're looking for. By the way, it is very sweet when you can get an older flagship receiver with all channels with preouts & main/ins, for a low price.
Cheers,
Bob