Hamas didn’t become a major player until 2006 when the Bush pushed for elections in Gaza, that the PA was supposed to win.
Hamas got the surprise win, and then the US and Israel funded and armed the PA/Fatah to try and overthrow Hamas, but lost.
In June 2006 Israel invaded Gaza. Then again in 2008 Israel invaded Gaza, and the final failed Bush foreign policy move was to force Israel to pull out in the last weeks of the Bush presidency.
^^ this isn’t a track record of supporting Hamas.
Israel did later allow money from Qatar into Gaza. The reasons are complicated. It started as part of terms of a ceasefire agreement. It was also a failed attempt to appease/moderate Gaza. Was there some idea it would keep the West Bank and Gaza divided, thwarting a two-state deal? Probably!! But Israel didn’t initiate payments as a strategy, it was more like a side effect of events.
Prime Minister Benjamin Netanyahu gambled that a strong Hamas (but not too strong) would keep the peace and reduce pressure for a Palestinian state.
During the 1970s, Israel began providing support to Ahmed Yassin, a Palestinian Muslim Brotherhood leader who controlled a network of Islamic schools, mosques, and clubs, in order to weaken the secular nationalist Palestine Liberation Organization.