I got the flush cut saw last time as my "decent" one needed a new blade, been brilliant for cutting off hardwood plugs and dowels etc. Lasted ages, still on the first blade. Can't fault it.
You could go for engineered oak (or you wood preference) glued down directly to the screed once it is at the required moisture content using a product like lumberjack 650 (I think it's called) or over laid on a moisture barrier sometimes with a cork type underlay.. Leaving the usual perimeter...