While this is true on the machine level, it also really depends on your programming language.
Certain languages like Python or Ruby automatically convert a smaller integer type to a larger, unbounded one. Other languages like Swift (or Rust in debug mode, I believe) prefer to throw an error upon integer overflow instead of silently wrapping. I believe that the silent wrapping behaviour in e.g. C and Java is a decision that prioritises performance over safety, as this is behaviour that you rarely want, and if you do, there should be better ways of expressing it.
Certain languages like Python or Ruby automatically convert a smaller integer type to a larger, unbounded one. Other languages like Swift (or Rust in debug mode, I believe) prefer to throw an error upon integer overflow instead of silently wrapping. I believe that the silent wrapping behaviour in e.g. C and Java is a decision that prioritises performance over safety, as this is behaviour that you rarely want, and if you do, there should be better ways of expressing it.