Why don't housekeeping genes have TATA Box regions in their promoter sequences?
Two possible answers:

First, TATA boxes has been found to cause increased "noise" in gene expression, such that the expression of genes that have a TATA box is highly variable. See this review for example.

A different set of results suggests (here) that TATA acts only as an amplifier of expression.

Regardless of whether TATA amplifies expression or also increases expression variance, this is not suited for housekeeping genes, which generally have low and constant expression.