It is not inconceivable that Israel might invite Egypt to do just that. There is always the worry, though that Egypt might not then leave.
“It is not inconceivable that Israel might invite Egypt to do just that (invade Gaza). There is always the worry, though that Egypt might not then leave.”
Two birds with one stone (defeating Hamas and outsourcing Palestinian population)- as long as Egypt withdraws its heavy armor back across Sinai. Only Egypt is big enough to absorb the Palestinian population without undue domestic political impacts. Al-Sisi governing Gaza would mean an Arab-style purge of Hamas/Muslim Brotherhood - ruthless, extensive, and largely ignored by the press.
In the future, the Egyptian Government might revert to a more inimical regime, but they would only be a bit closer than they are now in Sinai, and still bound by treaty to keep there heavy forces back on the other side of Sinai.