Is this good practice?
IMO that's a good approach, but make sure you are paying attention to what you're playing, and learning from the material: How the chords and melodies change in sound and construction, and the ways in which you have to alter your playing techniques across the different keys. Don't just play them off by rote. I practice a good deal like that - much more interesting than playing abstract exercises. (Which are also important, especially when you analyze and learn from them.)
But to develop your skills more deeply and more comprehensively, you might want to make sure that you are using a collection of songs that use different forms and types of progressions: 12 bar, 16 bar, 8 bar; I/IV/V , II/V/I; different types of bridge parts, etc. That way you learn how those various forms and progressions are played and how they work in the different keys.
From your selections it does appear that you have some variety, but maybe formalize it a bit: For example, make a list of songs you like to play (If you like to play something, you're much more likely to stick to your routine) that you know give you a decent array of forms and progressions. You'd probably be covered pretty well with 15 or 20 songs. Take four or five at time, for example, and play them in C. Then the next 5, until you get to the end of your list. Then move to G, or F, depending on if you want move in the sharp direction (clockwise) or the flat direction (counterclockwise).
If you select your material well, when you complete the whole cycle with all of your selections, you will feel really good about your accomplishment and will have learned a lot in the process. Then do it again - repeat a couple of times and then maybe try it in the opposite direction through the circle or some different order that you find interesting.
To complement the above, it's also a very good idea to play along with a rhythm track. It makes your practice sessions more interesting, more realistic and gives you more ways to challenge yourself: For example, keeping raising the tempo as you get more familiar with each key. Even if you'd never play the song that fast, you will develop your chops. Also experiment with different rhythms - any song can be worked over with any sort of rhythmic groove - try a jazz version of 'You are my sunshine', a samba version, an RnB version, etc. Sometimes it will sound good, other times not so good - work to make it sound as good as you can. (True - Amazing Grace with a samba style rhythm section at 130 BPM will probably never sound very good, but you never know - be adventurous and give it a try.)
With a rhythm track behind you, it will be also be easy to add a 'piano solo' verse to songs, where you improvise on the tune as if you were playing a solo - another way to keep it interesting, expand your knowledge and ideas, and help you when you're in a group setting, when very often (too often...) soloing a is big part of playing together.
Playing with others is always the best way to improve, but you can do great deal on your own if you learn how to challenge yourself with structure, discipline, milestones, goals, and of course experimentation.
If you take that approach for a while - 6 months might be a reasonable period - when you go to play with others you will feel confident that you are up to the task, and you may even find that you have the edge on a lot of other musicians.