It has recently been discovered that Stadia phone play won't work without Wi-Fi. This means if you were hoping to play your Stadia games on the train commute to work or home, you're out of luck.
At least for now. According to 9To5Google who acquired a confirmation from Google, Wi-Fi will only be required initially. That being said, there's no mention of when mobile data will work.
For the time being, Stadia may as well be considered an at-home-only service. Because you will need to be at a stationary location with a wireless router to take advantage.
Stadia phone play without mobile data support takes away a major benefit
One of the major, defining sales points of Stadia is playing on any screen. While this is technically a true statement, it comes with limits. You can play on any device you want that Google has already confirmed have support.
And you can kinda play anywhere, as long as there's Wi-Fi. Without mobile data support though, Stadia is losing a huge benefit it was supposed to have. Which is actually playing the games anywhere you like.
Instead, you're limited. Limited by the type of internet you use, regardless of whether your mobile data is fast enough. This might not be an issue for everyone.
But, there are no doubt users who were buying into Stadia for the ability to play these games while away from home and any sort of Wi-Fi connection. For now, those who wanted this feature will have to wait until it's rolled out. Which is god knows when.
Other similar services allow use with mobile data
While Google's reasoning here is to provide the best possible experience for Founders, similar cloud gaming services can offer that as well and still work on mobile data. Project xCloud works on a mobile data connection.
As does Shadow. One of those is currently in a private beta preview (xCloud), while the other can be readily paid for and used by anyone (shadow).
The point is that they both work on mobile data. So it's not an issue of whether it's possible or not.
For Google, it's likely about control. Just as it seems to be with every other limiting aspect of the service. It wants to ensure that it can provide the best experience to users early on.
And, limiting the number of compatible devices, as well as the internet connection type and compatibility for wirelessly connecting the controller is likely to provide some measure of control in helping Google achieve its goals of a smooth launch.