Actually, what you mentioned are not old and new pronunciations. Both are contemporary ones.
The "irohanihoheto..." is reading its letter names, while "irowa nioedo..." one, as a meaningful poem. The difference is much clearer if you try the same thing in English alphabet, say, "Cwm fjord bank glyphs vext quiz." (a perfect pangram from Wikipedia), where "iroha..." corresponds to "see-dublyoo-em-..." reading, and "irowa..." is "koom-fyord-...".
Each letter in Japanese syllabary was supposed to have the same reading as the syllable itself when it was created, but due to a series of phonological changes, the grammatically valid reading no longer matches the alphabetical one. As for lack of voiced consonants, it's said that today's voiced consonants are remnants of prenasalized sounds in Classical era, which they had no means to transcribe at that time.
The real "archaic" pronunciation around the 11th century would be like:
Irofa nifofendo tirinuruwo
Wanga yo tarenjo tunenaramu
Uwino okuyama kefu ko(y)ete
Ashaki yume mishi wefimo shenju
There are also some fun videos reading the Tale of Genji in Classical pronunciation.
What your "a modern version with updated hiragana" reminded me of was this (incorporating ん):
Which won the 1st prize of the "new pangram" contest held by a newspaper in 1903.