I have to disagree with your point about the references to modern technology sticking out like a sore thumb. I binge the whole show at least once a year and there are countless jokes throughout the series poking fun at technology that was relevant at air date. The first one that comes to mind is Amy’s comically small cell phone (which was a flip phone) that she couldn’t carry around because the wireless charger was the size of a toaster.

Another that I can quickly think of is the way they portray the internet, and the direct references to websites like Napster. They even resorted to making jokes based off of internet memes that have LONG since died off. For example, the entire conflict between Fry and Leela in the episode with the MomCorp eyePhone is based on the popularity of Susan Boyle’s performance on Britain’s Got Talent from who knows what year.

If references to modern tech are off-putting to you, you haven’t payed any attention in older episodes.