California has a requirement that any tax increase obtain support of two-thirds of the legislature. This has made California a complete fiscal basket case. Congressional Republicans have decided to import this model to the federal government:

Republicans have plenty of ways to push a constitutional amendment to balance the budget.
One option contains hard spending caps and a two-thirds supermajority vote for any tax increases. This version is embraced by the right but has little chance of passing.
Another option would drop those provisions, have roughly a dozen Democratic co-sponsors and stick to a straight majority rule for tax increases. This one actually would have a shot at getting the two-thirds vote needed to clear the House.
But nothing in this Congress comes easily, and Republicans have decided to go with the version that’s almost certain to fail but is beloved by the conservative base.

It's fortunate Republicans are so resistant to compromise, because the "moderate" version of the balanced budget amendment is also a terrible idea. The fixation on the two-thirds requirement is also pretty revealing. If you think the public agrees with you, why would you need a supermajority requirement? You don't see Democrats trying to impose a two-thirds requirement to cut Medicare or Social Security.

The answer, I suspect, is that Republicans understand that most Americans don't share their priorities. People may prefer lower taxes and smaller government in the abstract, but at the programmatic level they prefer actual government spending over low taxes. The danger of a balanced budget requirement is that it would force lawmakers to make specific, actual choices between competing priorities, and Republicans suspect, correctly, voters would choose higher taxes over slashing entitlements. hence the need to pair the balanced budget amendment with a requirement designed to thwart the majority will.