A new PC is no guarantee of it working TBH. iTunes is all DRM'd, and each update occasionally tightens it further.
Thus, point to point - app (iTunes) through every step to where it goes analogue again (generally the amp) have to support that version of DRM that the app is demanding. That includes the BT stack, the drivers, the cable, any mixer/graphic/DSP in the soundbar, plus any digital bits beyond. Even the standard audio driver in Windows can bugger it up, along with any codecs you might have installed.
And all have to support the standard and revision the app wants - which in the case of an unupgradable soundbar means replacement.
For HDMI, the copy protection scheme is HDCP from memory, and has a few revisions. No idea if iTunes uses that or something else, and nor do I care, as iTunes lives on one of my PCs out of necessity to upload to phone/ipod.
Honestly, I'd look towards less proprietary solutions - mine is CDs because I'm old fashioned and like to "own" it, and the amount of music coming out now that I want to listen to is limited. And I certainly buy less than £10 of CD's a month for myself. But if you like the streaming thing, other options are available without the Apple awkwardness. Cant personally recommend any, as not used any for ages. Mrs TB uses the free ad-sponsered Spotify occasionally, and I think Tunnie uses the premium version.
My own music tastes are quite varied, and for the most part, I'm happy with the wireless on for that variety, though not Radio 2 at lunchtime. But in the car if I'm in a hurry, I do did out the old favourites like Bat Out of Hell, or something like Canon in D Major if I need to slow down