In a conventional boiler setup, is hot water usually dispensed from the hot water tank quicker than it refills?
The tank fills at the same rate the water exits this is simple physics. However the tank starts heating water when some is removed. Since tanked water heaters are limited to about 5kw on average they have a limited ability to heat new fresh cold water. This is where tankless water heaters have an advantage they can supply hot water for long periods of time (I used to fill my 600 gallon hot tub at 103f every few months when I cleaned it. A 40 gallon tank well it has 40 gallons of much hotter water but in the US this is normally a 240v split phase circuit at 30 amps. The tankless electric can be 60, 90 or the largest I have installed 120a , this required a huge service upgrade to a true 400a service since the home was all electric and could supply 119 deg f water all day long where a tanked heater has ~20 minutes of hot water then the temp starts to drop because the heating elements only can supply 5Kw of heat but the pressure remains the same on a tanked system where a tankless system may reduce the flow to maintain the temperature.