the output of the serial console will be the thing to check but some system dont have a serial port anymore and most people dont have a nullmodem cable around (millennials might not even know what a nullmodem cable is without looking it up), you could even login the system this way (if it comes up its also possible that there is a kernel crash and the system freezes, you cant see the difference from looking at the video signal, that stops soon after booting)
that at least looks like the i915 driver was loaded as there are devices below /dev/dri
if you see this and you have network working your problem could be seen as solved
i cant do anything about the driver itself as i dont have the source (jun included newer i915 drivers in 1.04b but i dont have source to recompile them and i also not able to do a new backuprt of i915 drivers to the 4.4.59 kernel of dsm)
its kind of the same for the native driver from sysnology that comes with dsm
thats more or less the reasons why there are syno and std to choose, binary drivers from two partys we cant change
you can try to use the internal transcoding from synology with this
https://xpenology.com/forum/topic/24864-transcoding-without-a-valid-serial-number/
plex can have issues by itself on some systems, i doe remember some threads about specific problems with synology and plex where you had to delete some files from plex to make it work
i was about so say it should be possible to see this by looking at "drivers included (Reveal hidden contents)" in the 1st post here but looks like i did not look into that list for a while, i added 4 realtek nic's and now your r8125 is there too - so yes it should work, i did a new driver and if there is newer source and a newer chip/phy turrns up that need a newer version i will make a newer driver
beside my list you can also open the extra.lzma and look what *.ko files are in, you should see r8125.ko there too