ieee754 semantics aren't trivial despite a short spec. I implemented binary_float and binary_double for Oracle and did my best to make it ieee754 compliant (more so than the competition at the time) but there were still compromises:
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I thought that Oracle only had a handful of internal types (number + varchar?), with only a few hard-coded comparators needed for indexing. The rumor I'd heard was that these are mapped to user-visible datatypes in a linear way (e.g., timestamps are really number). Is that true?
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I am comfortable claiming that if you care about performance then you want datatypes for which memcmp is sufficient for comparisons (sorting, indexing, grouping).
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As disclosed in patent for the sort algorithm I added to Oracle, the new sort requires that. I am not sure if per-datatype representations have been explained, so I hesitate to share that.
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Does Postgres use memcmp for order/group by on a single column? Are the byte-comparable formats of the columns concatenated so that a single call to memcmp can be used? At this point, I don't remember the answer for MySQL.
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My observation is that getting "the prefix property" is the main reason to want memcmp()-able strings. IOW it's not so much about getting cheaper comparisons. The truly important thing is to *enable* optimizations that build on the prefix property, like your radix sort thing.