JKasten
The Ugreen 30985 ethernet adapter does use the ASIX AX88772C chipset and is the one that is currently linked to from the Roku Support page. https://oemdrivers.com/network-ugreen-30985
Although the product description on the Amazon page indicates it is not compatible with "any Roku devices", I think they mean to say not compatible (technically) with Roku Streaming Sticks. The recommendation/review section indicates users successfully using it with Express 4k devices.
Since it is a compatible chipset and users indicate it does work with Roku devices, you should be good to go with this device. 👍
I have used the Smays ethernet adapter with OTG (for usb storage) with my Roku devices and they work fine (with OTG, allows videos to be played through the Roku Media Player). With this setup, kind of an "improvised" Ultra setup by providing USB to the device.
https://www.amazon.com/Ethernet-Adapter-Replacement-Express-Streaming/dp/B09MVWZMMB/ref=sr_1_2?s=electronics&sr=1-2
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The Hitron en2251 is probably your sticking point. It is likely that you have only been issued one IP address. Thus, you can only use an unmanaged switch (or managed switch for that matter) with two wired devices if your ISP issues you two IP addresses (determined by MAC address) when directly connected to a modem.
It may be that the modem is having an issue providing internet access to both the MAC address of the ethernet adapter and the MAC address of the Roku device when directly connected to modem. (basically, a conflict is occuring as only one MAC address at a time can access internet with a single issued IP address.).
However, forgot to ask and just re-read that this is a "new" Express. Have you already setup/activated it? If not, try setting it up using a hotspot or other network before trying the wired method. May just need to get a LAN IP first (this will at least activate the device, but still don't think it will fix the issue you are having.).
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If you are adverse to wifi, then certainly placing a wired router (ethernet-only) after the modem, but before the switch will resolve the issues you are having. If not adverse to having wifi present, then likely any cheap router with the standard 4 ethernet/LAN ports will also resolve your issue.
Your modem has a 2.5G (multigig) ethernet port so depending on your speed plan and your budget, you can decide what type of router to purchase for your needs.
I can confidently tell you that a router (wired or wifi) will resolve the issue you are having and how I would address it. (you can also ask your ISP if they can upgrade your modem to include a combo modem/router if not adverse to wifi capability.) I would just purchase my own router, but laying out the options for you here.
Feel free to update or post more info if still having issues or questions.