Writing universal recursive function Is there a short explicit construction of an universal recursive function? All definitions I have seen involve numbering of Turing machines in some way, which is possible yet seems hard and unmanageable to write in a higher-level programming language (like Python, Haskell etc.)
Sure.  Write routines that compute each of Godel's primitive recursive functions, and write a routine for an unbounded search operator.  The set of functions you can compute this way is equivalent to the set of functions Turing machines can compute.  More information here.

The downside is that any input to your universal program would need to be framed in terms of those simple operations.  Of course, that is what compilers are for.