The government(s) will be happy to receive & evaluate your generous donation of upgrades to the myriad existing record keeping systems, and further generous donations of your time to train all staff on how to enter and search for the additional written characters, so long as the process does not significantly disrupt operations. "Patches welcome!" (^o^)/
But in seriousness: it is their concern because changes that seem minor can require major changes to electronic systems, and allowing special cases that necessitate lookup in ways the electronic systems can't handle (perhaps even requring manual search through hardcopy!) really throw a star-mangled spanner in the works.
Naming is already a special case problem in Japan, where people are accustomed to having no idea how to pronounce most people's names because the parents used non-standard readings of the kanji characters and/or used kanji from a special exempted list of archaic kanji that almost nobody can read. If you're wondering why the government doesn't require people use only the official "common use" kanji, at least just for this one problem, then I should tell it has been tried — enough people raised a fuss about being unable to register their child's "perfect name" that gradually a list of "allowed only in names" kanji was created and expanded.
And if I recall correctly, when registering a name, you can specify a totally unrelated pronunciation using the simpler non-kanji phonetic characters, so even just "common use" kanji are almost "all bets off". A relative few kanji have such common pronunciation in names that they can 'usually' be guessed.
And the problem is the same with many Japanese place-names, having little or no correlation between written form and pronunciation or meaning.
So, why is the spelling of a chosen name any concern to a government? It gets crazy out there, in Name Land. How mäný variatǐons of spelliñg cån ße ællowèd before people give up on pronouncing it?