Ok.
Where is that written? Your setting is your setting, but that's a homebrew setting and not a tightly woven expectation of the system. My setting assumes only that spells known by casters do not have to be strongly linked by theme. That is something different. How in the world can you say that D&D didn't expect spellcasters to have spells linked by theme? Bigby?? Otiluke???
Bigby, Otlieke, Tenser, and Leomund didn't have spells linked by theme. They promulgated spells they created in a theme. I'm pretty sure their books had a whole bunch of other spells in them and they didn't start with spells other than the spells known to their civilisations and almost certainly no spell on their initial list fit the themes these casters are now known for (that's why they invented what they did after all).
Diidn't I just prove how easy tighter themes and spell creation was in this trivial case? I think your justification for saying, "No.", here is pretty darn thin - and I say that as an advocate for DM's saying "No."
No not really. The value of a unique spell goes farther than the player feeling good that his PC is representing whatever theme the player wants. It exists inside the universe; it affects and informs the universe. The spells presented effectively represent the known "tech-level". New spells are a disrupting force, intellectual property that can be leveraged by those in the know in different ways.
In a universe where spell creation is hard and effects can only be generated if one knows the appropriate spell, the spell itself has potential value to other practitioners. Perhaps there are others who want spells in a similar theme. Perhaps they'd pay -- in cash or kind -- for access to this new ability. Perhaps this spell, now that it is known, can be used to make specific magic items more cheaply -- dropping the cost of a Frost Brand to rival that of a Flame Tongue, for example.