I think you are imposing an artificial distinction of your own making there, a Lughaidh

tá ... uaim can be either "need" or "want", i.e., it can be used for both.

The distinction isn't as important in Irish as in English (and the usage of these kinds of distinctions varies greatly amongst European languages.) "Want" originates from a meaning of "lacking" or "requiring" something. The Celtic languages seem to a

If you need to express "necessity" you can also use gá (dhom) or add teastáil to uait, but uait on its own can express "want", "need" or "missing" something.

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Maybe I was too quick the jump on the bandwagon!

No matter what you'd write here, if you pose it as a question someone will offer an alternative interpretation, which is good, although you put it in a perfect way yourself without the complications/ alternatives of others. Go with your own reading...it is good- as the alternatives also are.