so despite having ulterior methods of funding, hamas killing their opposition is because of Israel approved Qatari payments. despite the payments being made a **decade **after the violent takeover? did the money go back in time and change the election results?

While Hamas may not receive play by play instructions from Israel, it cannot be denied that they were allowed to > thrive to act as counterweight to more moderate elements

they were allowed to thrive because Israel left Gaza and let them sort it out. if you look back to 2005, you will see that israel (along other western countries) openly supported fatah over hamas. from wiki(emphasis mine):

Hamas leader Ismail Haniya formed a new PA government on 29 March 2006 comprising mostly Hamas members. Fatah and other factions had refused to join, especially as Hamas refused to accept the Quartet's conditions, such as recognition of Israel and earlier agreements. **As a result, a substantial part of the international community, especially Israel, the United States and European Union countries, refused to deal with the Hamas government and imposed sanctions **

compare to the PA in the same time(still wiki,emphasis mine):

According to the IISS, the June 2007 escalation was triggered by Hamas' conviction that the PA's Presidential Guard, loyal to Mahmoud Abbas, was being positioned to take control of Gaza. The US had helped build up the Presidential Guard to 3,500 men since August 2006. The US committed $59 million for training and non-lethal equipment for the Presidential Guard, and persuaded Arab allies to fund the purchase of further weapons. Israel, too, allowed light arms to flow to members of the Presidential Guard. Jordan and Egypt hosted at least two battalions for training

you can find many other instances of Israel supporting PA over hamas over the years. by the time bibi allowed funding to flow to hamas they had unquestioned military and civilian control in gaza.