I’m running multiple HDHomeRun ATSC DTV receivers at one location I maintain, with an older version of Elgato EyeTV in macOS, and encountering no issues. That setup is entirely wired, end-to-end.
First step here would be verifying that the HDHomeRun firmware is current.
Restart the HDHomeRun boxes, as well.
Then verifying the network configuration and the available bandwidth. Make sure the 100 MbE link is working.
If Wi-Fi is involved anywhere here (and your footer shows iPhone, which would involve Wi-Fi), verify that Wi-Fi link is working particularly.
Then characterize the issues and have a discussion directly with the folks knowledgeable in whichever app is stuttering.