Based on the product you linked and your aversion to "bedside computers", I selected alarms with a similar rear-projection type display.
Echo Dot (3rd gen) with clock - $45
Luckily for you, Amazon has an upcoming version of the Alexa-equipped Echo Dot that meets all the requirements. It's got a dimmable display, all the normal Alexa bells and whistles (including Bluetooth and Hue integration), and supports musical alarms with TuneIn, SiriusXM, and iHeartRadio (and if you have a change of heart and and decide to switch to to a streaming service, Amazon Music, Spotify, and Pandora.) Best of all, like all rest of the Echo line, it's quite cheap (as the hardware costs are subsidized by Amazon.) Unfortunately, they won't be back in stock until March 1st, 2020, but as long as you're willing to wait, it's a pretty solid box.
Sony S50G - $200
If you want better sound quality, prefer Google Assistant to Alexa, or just don't want to be quite so basic, Sony sells the SG50. Just like the Echo Dot, it's got Bluetooth, a dimmable display, and all the fancy bells and whistles that come with Google Assistant, including support for TuneIn, SiriusXM, and iHeartRadio (and again, Google Play Music, YouTube Music, Spotify, and Pandora.) Do note that there have been some (not super widespread?) reports of issues with internet radio on Google Assistant though.