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Yeah it's a conundrum, except for the fact that carrying around a DAC for my phone is so much worse than just having a music player. Having to pull out my little DAC dongle for 'serious' listening while I drink real ale at my seat in the pub. I'd feel dirty, like I'm carrying around a Fleshlight.
My phone has no external storage anyway, despite being a good phone (Xiaomi Mi 6). I could buy an LG Vxx phone, but I'd rather just not listen to music than dump a phone and spend money on another which is a bit louder. My last experience of streaming services was Tidal which entailed limited artists/albums and not being able to listen without wifi. Maybe that's changed, but I have a lot of the music I want on file anyway. The problem is that I can only listen in my bedroom. I have IEMs I like on that setup and 400 quid to burn on something more portable.
A DAP is the least annoying option to me, but whether it worth spending a few hundred quid on I'm not sure.
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I too have a reasonably large library of lossless music and am not really that into streaming, though I accept it does have its place.
My solution for almost a decade now has been a NAS at home that serves my music to my phone/work computer over the internet using Plex. Plex essentially creates a private streaming service that can adjust the bitrate of the stream depending on whether you’re connecting over LAN, WiFi or mobile data. Each device has its own DAC with the quality Of the DAC dependent on use.You could certainly do this for £400 and a bit of time invested, especially if you used a micro PC with some storage rather than a NAS. If your files are already on an internet connected PC / Mac then you could do it for free or the cost of a plex licence / subscription. Plex does have limitations though, for example it won’t support MQA.
Sorry, I know this doesn’t answer your question but it does give you an alternative option :)
Unless you’re already familiar with using a DAP I think it’ll probably end up becoming annoying having two players. Why not upgrade the phone’s storage (or subscribe to some streaming music service or other) and add an inline DAC like the Audioquest dragonfly or Cyrus Soundkey ?