And why didn't Merchan just put the case on hold until Jan 20, 2029 when Trump becomes a private citizen again?

I imagine it's because that's utterly unprecedented. It might specifically violate some provision of the NY criminal rules of procedure, which probably don't contemplate a four year delay between verdict and sentencing. But even if it doesn't, that's something that probably has never happened before.

Criminal defendants are entitled to due process - and while that's in some ways a pretty vague and nebulous term, it generally encompasses the right to have a criminal case processed under normal and regular rules. It would be highly unusual for a judge to delay sentencing for four years, and especially to do so for the purpose of being able to have more power to impose a harsher sentence than would be lawful at the moment (to phrase it in a way Trump's lawyers might). That might pose a defect that prevents sentencing at all.

I'd also heard that postponing the case would allow Trump's lawyers to have 4 years to come up with a way to get the case thrown out, but they're going to be doing that anyway, aren't they?

Not really. There are a few legal disabilities that come with being a convicted felon that don't inhere to someone who had their case thrown out, even if no additional penalty is imposed after the verdict. And there are obviously a lot of non-legal, reputational consequences that come from being a felon than someone who was never convicted.