Two different concepts at play here.
first is the function of the flywheel, it's an inertial body, intended to smooth out torque pulses...it does this by storing energy (aka flywheel energy storage systems that are being discussed ATM).
Take a 4 stroke engine, single cylinder, and there needed to be something that carried it through the exhaust, intake and compression strokes. The flywheel gets accelerated during the power stroke and drives the engine through the balance...
Extreme version seen here in the hit and miss engine...still capable of power delivery through the action of the flywheel even when not firing.
If you've got a massive flywheel, you will have an engine that uses some of it's instantaneous power output to accelerate the mass rather than the car, but you will also have more "apparent" power available as you let out the clutch, as the rotational inertia of the flywheel will add some apparent power as it is slowed.
That's why light flywheeled cars feel more responsive (they are) and are more quirky off the line.
Note in the above video that the flywheel's mass is all located at the periphery of the flywheel.
The inertial effect of any mass is related to it's radius of gyration to the power of 4...per pound you get 16 times as much flywheel effect at 2' radius than you get at 1' radius.
When you see a lightened flywheel, the fly cuts are at the periphery, between the clutch pressure plate bolt locations....where the lest mass has the most effect.
There's little sense trying to achieve the same effect internal to the engine, as to get the same inertial effect, you need to put in so much additional metal...the stuff in the crankcase is not intended to be a "flywheel", but DOES contribute an amount.
As OVERKILL has demonstrated, and engine can be built without counterweights, and it will run.
Counterweights are intended to balance out vibrations due to rotating ad reciprocating components, and can range from none, to full, to "overbalanced".
Here's the evolution of the Holden 6 cylinder.
First is what they originally came with, one counterweight per crank throw.
And how they ended up, with "full" counterweights on the later engines.
By having the piston and rod masses balanced within the vicinity of the crankpin, i.e. between the two main bearings, the inertial loads don't make it to the main bearings, making their life a little easier. In the earlier crank, you should be able to envisage that the one sided counterwights will have a little more crank flex, and uneven main bearing loading.
http://www.bobistheoilguy.com/forums/ubbthreads.php/topics/4002532/Harrop's_Howler
He cut them off entirely to get max HP on a drag strip.
In the extreme, a crank can be built that has no counterweights, and is externally balanced either through the crank forging, or weights in the harmonic balancer or flywheel.
As a driver, you would never know whether the crank in your engine was internally balanced, externally balanced, or otherwise.
90 degree twins can be perfectly balanced by balancing to a bobweight on the crankpin of all the rotating parts, and 50% of the reciprocating mass...(balancing to more weight is "overbalance).
Balance shafts are becoming common these days, and it's not to do with counterweights on cranks, it's because 4 cylinder engines in particular have problems with a 2xRPM vibtarion which is caused by the action of the crank and rods not having the motion of the piston as a true sinusoidal wave...pistong spends a lot longer around BDC than at TDC, and the balance shafts are timed to knock that out.
You'll also hear "dual mass" flywheels particularly on 4 cyl diesels these days. These are to damp out torsional harmonics, just like a "harmonic balancer" on the front of the crank.