DisplayPort won everything, except not becoming the physical connector for home cinema. Heck, even within those HDMI-exposing devices, DP won.
The vast majority of display drivers speak eDP. Few things actually implement HDMI, and instead rely on DisplayPort to HDMI converters - that's true whether you're looking at a Nintendo Switch or your laptop. Heck, there is no support for HDMI over USB-C - every USB-C to HDMI cable/adapter embeds a HDMI converter chip, as HDMI altmode was abandoned early on.
The only devices I know of with "native" HDMI are the specialized TV and AV receiver SoCs. The rest is DP because no one cares about HDMI.
However, seeing that home cinema is pretty much purely an enthusiast thing these days (the casual user won't plug anything into their smart TV), I wonder if there's a chance of salvation here. The only real thing holding onto DisplayPort is eARC and some minor CEC features for AV receiver/soundbar use. Introducing some dedicated audio port would not only be a huge upgrade (some successor to toslink with more bandwidth and remote control support), but would also remove the pressure to use HDMI.
With that out of the way, the strongest market force there is - profitability - would automatically drive DisplayPort adoption in home cinema, as manufacturers could save not only converter chips, but HDMI royalties too.