"All I'm trying to saying is that if the Executive in a democracy gets voted in after campaigning / about a certain policy, then is it not rather undemocratic for the House to block / prevent / refuse funding for the policy?"
The individuals we (including the US electorate) elect have their own responsibilities and that includes holding the executive to account. That includes preventing them from undertaking something wildly impractical or harmful and maintaining a control on the purse strings. In fact, it's the latter which is one of the main tools they have to do that.
They have their own democratic mandate to do this.