How To Send TuneIn Radio On Android Device To A/V Receiver?

I can't vouch for any of them, but there are dozens of apps that provide Airplay or DLNA functionality. Search the Play Store for "Airplay" or "DLNA".

I'm sure you want to take advantage of the built-in Airplay/DNLA functionality of your receiver but one of the most convenient solutions is simply to add a Bluetooth receiver dongle to your stereo system. They're only ~$10 - $17, depending on whether you need the USB charger and audio cables. Works great. Sounds good. Very portable in case you want to take it with you to wear it, or take it to a vacation house stereo, hotel stereo, etc.

Bluetooth A2DP 3 5mm Stereo Audio Receiver Dongle Adapter for PC Speaker Phones | eBay

Wireless Bluetooth A2DP 3 5mm Stereo HiFi Audio Dongle Adapter Receiver T9 | eBay

If you want to send a digital (SPDIF) coaxial or Toslink/optical signal into your receiver (for higher fidelity) there are bigger, more expensive ($35) Bluetooth receivers like this...
http://www.ebay.com/itm/Bluetooth-4-0-Music-APT-X-Audio-Stereo-Wireless-Receiver-for-iphone-mp3-hk-/400667645582?pt=US_USB_Bluetooth_Adapters_Dongles&hash=item5d49a7168e

Bluetooth is just so easy and has so little lag. It should be built-into all new receivers.

 

Unfortunately you are not going to find a solution that does not require root for the scenario as presented, primarily because of copyright/licensing/Fair Use restrictions.

The root apps can hook the raw audio as you already stated, any apps that stream DNLA only do so from approved apps or content you store locally thereby assuming you have fair use rights to stream that content like videos and music.

You can brush up on rooting here, with a video - How to Root the Samsung Galaxy Note 3 – XDA Developer TV – xda-developers

The original thread - [SM-N900*] CF-Auto-Root - xda-developers

Pay special attention to the multitouch issue at the top of the original thread.

 

I have 2 houses (one is a single story, and one is a 2-story). In both houses I can stream music to the cheap Bluetooth receiver (in the Ebay auction links above) from my phone to my stereo (through multiple walls) from every room in the house and from the patio or back yard. Maybe I'm just "fortunate" to have small enough houses. :-( But in both cases the stereo is located on one end of the house (not centrally located). Maybe your house's layout is less conducive or maybe your old Logitech device was weak. You probably already thought of this, but you may want to consider moving the Bluetooth receiver to a location (e.g. on top of your stereo armoire) such that the radio waves have an easier path (without passing through metal or cabinet doors, or at oblique angles through walls to your patio. Good luck!