If I have grasped it correctly, the capacitors on the power-pins are required to remove
the DC ripple effect.
No, the power supply capacitors are for decoupling - for logic circuits they prevent the high-speed(*) switching transients from all the 10,000's of transistors adding up on the supply rail and causing circuit misbehaviour - incorrect logic switching. Since the Arduino chip has analog inputs as well the decoupling also is very important in reducing analog noise. With high speed decoupling its important the capacitors are right next to the chips connected via a low impedance (low inductance) path - that means short fat wires/PCB-traces.
Often voltage regulators require a certain amount of capacitance on the output rail to be able to live up to their specifications. Typically you want 0.1uF ceramic caps next to each chip and 10uF to 100uF somewhere on the board for lower-frequency decoupling.
Inadequate decoupling shows up as unreliable behaviour, pattern sensitive and difficult to trace with a logic analyser or 'scope since monitoring the signals changes the high-speed transients that are causing the problem. It will drive you crazy trying to work out what's happening, so never skimp on decoupling.
Incidentally DC ripple should be handled by the voltage regulator circuit/chip.
(*) modern CMOS devices switch output pins in 1 to 10 nanoseconds, but internally some might be switching in 10's to 100's of picoseconds (and have local decoupling capacitors built onto the surface of the chip!).