If anything I've got to say the end result your setup is producing sounds great to me, I prefer to hear a more bassy and not so heavily mid-to-high audio signal when I monitor comms myself so that sounds great. Since you say you have a 996XT which can do P25 Phase I my next question is whether or not this is supposed to be a digital system because if you didn't mix up the audio sources then what I'm hearing from the first section of the video (the 996XT I presume as stated) sounds more like a P25 Phase I data stream being decoded, at least that's what it sounds like to me given I've used Unitrunker with DSD+ to monitor some P25 Phase I traffic in my area and that's pretty much what I get as a result myself.
The second audio portion of the video, which is stated as being the RTL based reception, sounds more like what I'd expect from the 996XT myself - in all my years I've only had a few minutes of experience using an actual digital scanner (a 436HP I got to play with for a short period a few months ago) and when I plugged in my Koss KSC-75 earclips that's about how it sounded to me decoding a P25 Phase I stream here in the Las Vegas area (there's not really that much on P25 Phase I at the moment, 3-4 systems overall but DMR is catching on really fast here).
I'm not saying you made a booboo and if you're actually getting audio that clean and clear from your Unitrunker based setup with an RTL stick my advice is this:
don't change a thing. 
Aside from that, I'd be interested in knowing the setup overall you're using - the only time I get audio that sounds similar to yours in the video is by using SDR-Radio aka SDR-Console because of how Simon (the software's author) uses some nice filtering in the audio pathway and it produces great sounding output - again, to my ears it sounds like I'd want voice audio to sound, not everyone has the same ears and has different expectations and requirements obviously.
But definitely, spill the beans 'cause it sounds great.
If it's a pure analog trunked system then everything I just said is moot and means nothing at all since for myself personally there's never been and probably never will be a digital comm system based on radio technology that's going to sound like pure analog does but that's also my opinion. I know there's been a serious effort to produce understandable speech using digital codecs but they still suck in major respects, every one of 'em.
