Well, if you spent a ton of money; to have all that work done, you might take the car back to them. It maybe a unrelated issue or not; they should check it out, fix it for free or give you a estimate on the repair, then you can go from there. They jacked with it last; let them deal with it first.
How is the car running otherwise (any check engine lights or trouble codes); just for the heck of it, I would check the timing marks and make sure, they got the timing right... when, they swapped the belt. If it is a tooth off or so; it will usually throw a trouble code.
Was looking at the video, you mentioned that the first week everything was fine. I do not believe it has anything to do with the timing belt. I would look closer at the drive belt and your alternator, it appears to me that your battery is not fully charging and as time goes on it will get worse on starting. Not enough cranking power.