@knottyrope, up to you, but I will only thank tsac when they stop trying to suck all positive energy out of the web!
I found a proper fix: "simply" patch the application that displays the prompt.
Many third-party applications contain a string that they use to check that a "Play Services compiled by Google" is installed on your phone. Chances are that if you have Play Services on a BB10 then you are not using the Play Services compiled by Google. (You are probably using a version modified by Cobalt232 on the dark side).
The "Missing Google Play Services" prompt means that the two strings above did not match, even if the binary is identical.
How do you fix that? You could change the string in the app (executable/binary) using a HEX editor, which is not hard but it is tedious. You could use the "Play Services Patcher" developed by Cobalt232 for Windows to update the binary for you. Use Google to find your preferred tool.
This kind of "patching" might also fix apps that prompt and then immediately exit.
These fixes are fairly straight forward because BB10 is a variant of Android, just like MacOS is a variant of BSD. Some "tech news" sites claiming that Android apps cannot work with 1:1 screen ratio are baloney. Other "trusted publishers" make erroneous claims that apps on BB10 can only use the pre-installed BlackBerry services for tasks such as mapping. As a rule of thumb, do not trust hasty Internet posts! The reality for me is Android apps can generally be made to work on a "Classic" with the caveat that the GUI proportions might result in a sub-optimal experience.
I have rock solid stable BB10 OS, a battery that lasts 48 hours between charges (I leave it permanently in low power mode), and all the apps I need; so hopefully will not be tempted to turn off the phone. The phone is, however, somewhat slower than I would like it to be.