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private_messaging comments on Preferences without Existence - Less Wrong Discussion
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Should not be a problem for you to name one where it doesn't work, right?
Universal TMs all have to permit setting up an interpreter for running the code for another UTM, you really have a lot less lee-way for "methods of specification" than you think.
If you are hung up on there being an actual difference in the output, rather than hand-waving about the special circumstances or such, you ought to provide at least a sketch of a set up (so that people can understand clearly how you get more of the simpler programs). For example, for programs that do halt, instead of halting you can make them copy extra bits from the input tape to the output tape - that would be really easy to set up. And the programs in general can be compactly made to halt after something like BusyBeaver(10) steps.
Whether those L+1, L+2, etc. count as different programs from the length-L one is, last I checked, contentious, and because theorists feel strongly about this, various different widely-used formalisms for defining what makes something a UTM disagree about whether unread tape matters. If someone pokes a hole in the argument in the great-grandparent so that the general conclusion works iff the 2 L+1, 4 L+2, etc. count as different universes, then it would become worth addressing. But as long as it works without it, I'm going to stick with the most difficult case.
Again, give an example for the assertions being made.
As for your argument, as others have pointed out, you did not prove anything about the extra length for setting up special output in very specific circumstances, or sketched how that could be accomplished. "Sticking with the most difficult case" you got into the region where you are unable to actually produce an argument. It is far less than obviously true (and may well be false) that the programs which are simple programs but with special output in very specific circumstances, are a notable fraction of large programs.