22/7 is a marginally better estimate than 3.14 (~.04% vs ~.05%), both at the "cost" of remembering 3 digits. Are you suggesting that it's easier to remember "." than "/"? Not sure where you're going with this.
Both have, more or less, the same the difficulty to remember. The "22/7'" disadvantage is that you still have to perform the division to have the value while 3.14 already is the number.
I remember out to 3.14159, but that's wasted space and I'm not sure how those wasted digits snuck in there. Either 3-digit estimate is good enough for just about everything you're likely to run into.
I remember also 3.14159.
Of course, when you are in a situation where you need more digits. (I faced moments like that...) you probably already have the tools to fetch an approximation with more digits...