In your example sentences, "need" is countable, not uncountable.
This school has many urgent needs. There's an urgent need for qualified teachers. There's an urgent need for new classrooms.
To fulfill all emotional needs. To fulfill an emotional need.
An uncountable use is:
This school is in need of qualified teachers.
It doesn't make sense for a noun to be both singular and uncountable at the same time. Oxford Learner's Dictionaries says it's "singular, uncountable". That means it's singular or uncountable not singular and uncountable. Cambridge Dictionary is clearer saying it's "S or U" (singular or uncountable (not both)), with other definitions being "plural"; "C or U" (countable or uncountable); or just "U" (uncountable).
- Oxford Learner's Dictionaries: need
- Wiktionary: need
- Cambridge Dictionary: need
So your question is based on a fallacy. Uncountable nouns don't take "a/an". If it has "a/an", it's a countable noun.