If you combine capital=yes it with place=city and a certain population number (>500.000?) you can filter all less important capitals (for instance those of smaller countries/ mini nations, states and provinces or even lower districts) for the lower zoomlevels. You can also look if those capitals of lower admin. districts are part of a relation lower < admin_level=2 to filter them out. So this way I think you can avoid that capital cities / towns that are either too small or not a capital of a nation are rendered too early.
Hi, capital=yes is also used for regional capitals, eg. capital of a county or lower administrative areas. Note: there are only 193 countries accepted by UN and additional 13 ones not accepted. So I would recommend to use is_capital=country in combination with place=city.