Of course, it wouldn't be 'sportswashing' with them. It'd be about an upwardly mobile, developing country bringing some pride back to a historic & neglected club & restoring it to its rightful place.
And investing in Liverpool, a city that has been starved of investment by successive governments since Thatcher. (apart from getting billions in Objective One funding from the EU, along with billions in investment from successive governments since Thatcher) I bet Liverpool Council which is under government supervision and is being run by councillors and ex-councillors from other cities can’t wait for the Qataris to but LFC and to invest money in Liverpool.