No, they do not. Tax exempt organizations shouldn't be private and also exempt from government oversight. I do think that legitimate small businesses should be tax exempt to simplify beaucracy, though.
I think that businesses below 200k annual income are probably not worth taxing outside of income taxes on employees. It'd probably spur small business entrepreneurship which I think is pretty healthy for market competition.
Then a government would be incentivised to write policy to benefit bigger organisations and hurt small businesses. The government would make more tax revenue if they encouraged the bigger, taxable, orgs to crush the small business and absorb their previously non-taxable market share.