The 2nd HDMI out on the 2300 is a mirror of the first, therefore not capable of running a separate source from the first. You'd need a 4200 or better for that I believe.
The 2300 would be a step or two down in terms of sound quality as well I think. Whilst being feature packed, it still is an entry level receiver.
What speakers do you run, OP?
Unless you want Dolby Atmos/DTS-X (You need ceiling speakers or speakers that direct sound upwards from the listening height in addition to what you already have AFAIK) Atmos blu-ray/Ultra HD software is growing all the time right now. DTS-X is a bit slow getting off the ground. 4K passthrough is something you may also need.
THX (if you want that still) You may be looking at another Onkyo.
Either way an upgrade with the above features & a step-up in sound quality isn't going to be cheap.
Budget, such that we don't suggest a Trinnov?
Hmm at that price range you're really not going to be upgrading in terms of sound quality, just features.
You really need a budget of 1k+ to get a worthwhile upgrade.
For Atmos you either need a pair or more of ceiling speakers or upfiring speakers which sit on your front LR.
Not sure about streaming as I prefer physical media.
If you're not buying into 4k then upgrading to recent avrs is pointless. If you do want to upgrade then you need to be looking at high end avrs from the pre 4k era. Something that cost 5k 10 years ago is still a high end avr sound quality wise. The only issue then is the lack of more modern audio features.