[Dr.Feelgood]

Memory should be taken out of your equation, as the J4205-ITX doesn't come with it either.
The below are all lowest prices in my country.
The J4205-ITX costs 126.51€ for me.
For 100.50€ I've got a new 1151 board (56.30€) and a Kaby Lake Celeron (44,20€).
So there's a price difference of 26€.
An RX 460 can be had for 111€.
Meaning you'd save 85€ going for the J4205-ITX.
Not to forget that DDR4 is generally less expensive than DDR3 SODIMM. Haven't got the time to scour the market, but the price difference between DDR4 and DRR3 SODIMM is quite large by the looks of it. Mainly because DDR3 SODIMM stock is running very dry.

Long story short: a system with a separate GPU, CPU and motherboard is at the end of the day costing quite close to the J4205-ITX. And the difference in power efficiency is not significant. Sure on paper it is significant if you run it 24/7/365, but if you're using it as a media player, and only for the occasional movie or TV episode, it won't run enough hours a day for me to let the difference in power efficiency be a relevant factor.

Edit:
But since the J4205-ITX along with 8GB of expensive DDR3 SODIMM is already in my possession, I'll try to RMA it and in return hope to get a proper functioning board like yours. But at this point, it feels like a lottery (which it most likely is).

I agree that there are many different options and there is no "one size fits all" solution and no "black or white" statement with reference to working or not working.
However as this tread is about Apollo Lake i would say that as platform Apollo Lake is very good solution. It's fanless, low power consumption, manages all video/audio formats with ~25% CPU usage, have both Linux, Android and Windows support and is delivered by Intel. As such it will only grow in applications and support for the future.
Same goes for Kaby Lake Celeron/Pentiums but unfortunately currently there are no B250 or H270 motherboards with HDMI 2.0, while the Z170/270 HDMI 2.0 boards are way more expensive than alternatives. As soon as there is cheap B250 chipset motherboard with HDMI 2.0 you can scrap the separate GPU as there will be no need for it anymore ...