I haven’t found a good answer myself, but I also want desktop syncing. I have tried a lot of them, but everything falls short of my needs.
I do my budget in Excel but use Banktivity on my desktop for tracking my spending, which has an iOS app. I don’t pay for bank syncing, I do it all manually. Banktivity has budgeting, but I really do not like how they do it.
I would say You Need a Budget, easily the king of budgeting, but I dropped them once they went web based with a monthly fee.