In the years since this question was asked, there are some new technologies that offer better solutions.
Hammerhead Karoo 2
For a completely smartphone-free bike GPS experience, Hammerhead's Karoo 2 is your best bet. It doesn't even have a smartphone companion app, and can run entirely off WiFi (for data syncing only in WiFi range) or use an installed SIM card for an always-on cellular connection with live tracking during your rides.
At $399 it's pricy, but not far off less-capable options from Garmin and Wahoo.
It's regularly updated (more frequently than Garmin or Wahoo's devices), but as of this moment one of its missing features is that it'll only let you use Karoo's live-tracking app, and doesn't integrate with Strava's live track service. The key difference is that Strava's service will message your contacts when you start and finish a ride, and show your progress during the ride, whereas Karoo's live tracking is just a map that shows your current location, and won't allow you to auto-notify anyone, or show your progress for your current ride.
Wahoo ELEMENT Roam & Bolt
If you're not worried about live tracking without a smartphone during your rides, Wahoo's bike computers are the other option for riding without the need to connect to a smartphone during, or after your ride.
From a features standpoint, they're similar to Garmin's Edge line of computers, but the Wahoo ELEMNT Roam & Bolt both have WiFi to upload rides to Strava, and other services without syncing to a companion app.
You'll still need to pair an iOS or Android device for the initial setup, but after that, you can largely forget about it.
There are a few features you'll miss if you leave your smartphone at home during your ride, like live tracking, but they're fairly minor.
The other options are WiFi enabled smartwatches. For example, Apple's Watch devices with cellular will allow you to use Strava's Apple Watch app to record and live-track your rides while leaving your smartphone at home. (Though, notably, Strava's app won't work with "Family Setup", which allows you to setup an Apple Watch without linking it to an iPhone).
Other smartwatches from Garmin and other manufacturers have similar functionality. However, I get the impression that the original question was looking for bike computers specifically, not wrist-mounted devices.