Never, in any engineering field, have I EVER seen anyone simplify pi to 5. For that matter, I have never seen anyone simplify to 3. It is always 3.14. I feel like pi simplification is a weird meme that people think engineers do but is never practiced anywhere.
It’s like if there was a meme about chefs saying they always replace eggs with grapefruit. No they don’t, and it’s nonsense to think they do.
The only time you should be doing head calcs as an engineer is to double check that you have a reasonable answer with the actual calcs on your actual calculator.