According to most economists, is it a serious shortcoming of GDP that it does not count household production or production in the underground economy?

No. "Household production" does not contribute much to the GDP.

Explanation:

To arrive at the reason, first we need to understand the term "GDP" and "shortcoming"

Shortcoming means failure to meet certain expectations.

GDP: It is the snapshot of the economy which speaks about the production in certain period.

Since the household production does not come under market transaction and since GDP involves only market transaction, household production cannot be considered and thus it is not a serious reason for shortcoming of GDP.