Is there any modern video connector you can use royalties free?
For devices and endpoints the contemporary digital interconnects have an ensnarement with HDCP. This includes both HDMI and DisplayPort, which can support driving HDMI with an ‘active cable’ adapter.

Besides HDCP, HDMI has a ‘Logo License’ that costs serious money. DP doesn’t, though MPEG LA wants to charge a (disputed) $0.20 per unit royalty fee. 

DVI doesn’t have an HDCP requirement, it’s optional, and it has no fees. This might be a reasonable choice that’s still supported in the marketplace, and it’s supported as a legacy mode in HDMI with a passive adapter.  (Yes I’m aware of TMDS patents. As a practical matter those are rolled up in the source/sink device, not the connector itself.)

An HDMI source however can choose not to send DRM-enforced content over DVI, or it can down-res it to 480p. This applies to any device that it can’t identify as HDCP compliant, including DisplayPort that isn’t HDCP enabled.

Where would this come up? Pretty much any playback of copyrighted material, including streaming and packaged media.

DRM sucks. And I say that as someone whose signature is on an HDCP license.