WIRED CarPlay issues when using Cloudflare 1.1.1.1 -- why should it matter?

Putting the question first: why would the functionality of a WIRED CarPlay connection -- with the car playing songs saved on the tethered iPhone, not streaming, and WiFi turned off -- be affected by the performance of a "VPN", in this case the Cloudflare 1.1.1.1 app?

This makes absolutely no sense to me.

The scenario

Have used a wired-only CarPlay connection in my car since April without issues, including running 1.1.1.1 on a iPhone Pro Max 15 with WiFi turned off (iOS versions kept up to date).

This week, I applied a software update from the car maker to gain wireless CarPlay (and other) functionality. My first post-update drive using wireless CarPlay was a mess: dropped audio in the car (when the streamed Apple Music song appeared to continue playing in Music on the phone), random disconnections, graphics glitches, etc. Some of these issues may simply be bugs in immature software that the automaker needs to squash.

HOWEVER, while parked in an area with poor cell signal and playing songs saved to the phone, no longer streaming -- and after deleting the initial car and phone profiles, turning off WiFi, and switching to WIRED CarPlay -- I again experienced audio dropouts that correlated with the re-appearance of the "VPN" icon next to the battery indicator on the iPhone (as if Cloudflare had stopped working owing to signal issues and was reconnecting).

I experienced no further audio dropouts on the wired connection after disabling 1.1.1.1

Why would interruptions to the 1.1.1.1 service / "VPN" connection affect WIRED CarPlay-cast audio from songs saved on the device?

Adding, I do not recall any such audio playback issues (and I presumably drove through similar areas of poor cell signal) when using WIRED CarPlay and 1.1.1.1 prior to the car software update.