No it's not, you can write an extremely large set of programs in non-Turing complete languages. I personally consider Turing completeness a bug, not a feature, it simply means "I cannot prove the halting problem of this language" which blocks tons of useful properties.
You can write useful programs in Agda. I wrote all kinds of programs in Agda including parsers and compilers.
You can write useful programs in Agda. I wrote all kinds of programs in Agda including parsers and compilers.