In fact, the House does control the purse strings. The only question is whether they are willing accept the consequences of their choices.
The House could set aside any budget proposal from Obama and craft their own budget and pass it. They have no obligation to consider anything Obama submits.
The rest of the government may not like it. The Senate may not pass it, and Obama may not sign it, but then at least it's clearly their choice not to fund the government.
The mistake the Repblican House is making is accepting Obama's proposal as the baseline, to which anything less is a "cut". The Constitution gives no budgeting role to the President.
They aren’t willing to accept the consequences because they have no guarantee on how that would end up.
The democrats did that with Obamacare and lost a whole lot in 2010. That happened with a pliant and willing media to support them in every way. The GOP does not have that.
That is one of the reasons they are skittish.