The main difference with an un-pressurized tank with a copper coil is you only have one pumped flow, so the coil sits in a "pool" of warmed water. Only the thermal induced movement in the tank creates the driving ∆ T.
With a pressurized system you have only one HX and that is the boiler itself, the tank or load flow goes directly to the HX and it is a moving flow against the HX.
That the essence of a plate stye HX, you have two pumped flows and you have a much more efficient heat exchange. It is possible to size a plate style HX (close approach) to within a few degrees, A side to B side. It amazes me how much energy a 5X12" 30 plate HX will transfer. I generate my DHW instantly with a single pass through a 30 plate HX. 120F boiler side, 55F incoming water gives that huge ∆ T
Adding fins, or more surface area to the smooth copper coils would speed things up, but moving the flow around as well as through the coil would be a big plus. Many tank manufacturers now use the corrugated stainless coils, more surface area, plus the rough inside surface encourage additional HX via turbulance. Just like the turbulators you add in the boiler flue gas passages.
E dub is correct a BTU is BTU. If the boiler is firing then the energy is going somewhere. If the boiler goes into idle mode with a warm tank, then you lack heat exchange efficiency. If it has a mixing valve the operating temperature rating PLUS the differential needs to be combined. A 140 F element with a common 18 F differential = 158 F to the return before the bypass shuts down. Not all thermostatic valves have a 100% bypass shut down, so you are always slipping the clutch so to speak. I just dropped my thermostatic sensor cartridge to a 130 F. I may drop to a 115F.
Remember the dewpoint with wood varies greatly depending on the moisture content. It's not like gas, oil, or LP with a predictable dewpoint and 130- 140 F is realistic. With wood it is a balance more air to the combustion process lowers the dewpoint, but more air also carries more energy up the flue
Heat goes to cold always, but also the rate of hot to cold depends on the differential. The large cold start firings show quick temperature rise, as the load or tank warms the exchange slows. That is where a more efficient HX kicks in. Seems those last 5- 10 degrees are painfully slow to transfer, especially when you are watching and expecting.
http://www.caleffi.us/en_US/caleffi/Details/Magazines/pdf/idronics_10_us.pdf