I know persistent lava lakes are rare, but do they occur in all volcanoes originally, and just not persist over time?
No, they do not.
Lava lakes require the lava to have low viscosity (that is, be very fluid or water like). If you look at the Wikipedia article for Lava lake, you will see that "basalt" or "phonolite" are mentioned. Those are indeed types of lavas (actually rock) that are very fluid. These are the kind of lava that end up forming lava flows.
On the other hand, there are other types of lavas that do not flow, because their viscosity is too high: rhyolite is one example. Explosive volcanoes (think Mt St Helens) are explosive (mostly) because of the high viscosity of the lava. Therefore, volcanoes erupting explosively (almost) never have lava lakes, because the lava itself is not fluid enough to make a lake.
A great many volcanoes have really deep magma chambers. At the surface they eject ash clouds or pyroclastics. There are also agglomerate type volcanoes, the deepest of which are kimberlite pipes. So no, don't expect to see lava lakes in many volcanoes.