Why does Li-ion charging require very tight voltage tolerance?
Overcharging

Excess charging voltage leads to plating of metallic Lithium. Damage. This can lead to thermal runaway (vigorous fire)

Your link to Battery University explains why the charging voltage should not exceed 4.20 V / cell


  Overcharging Lithium-ion
  
  Lithium-ion operates safely within the designated operating voltages; however, the battery becomes unstable if inadvertently charged to a higher than specified voltage. Prolonged charging above 4.30V forms plating of metallic lithium on the anode, while the cathode material becomes an oxidizing agent, loses stability and produces carbon dioxide (CO2). The cell pressure rises, and if charging is allowed to continue the current interrupt device (CID) responsible for cell safety disconnects the current at 1,380kPa (200psi).
  Should the pressure rise further, a safety membrane bursts open at 3,450kPa (500psi) and the cell might eventually vent with flame. The thermal runaway moves lower when the battery is fully charged; for Li-cobalt this threshold is between 130–150C°C (266–302°F), nickel-manganese-cobalt (NMC) is 170–180°C (338–356°F), and manganese is 250°C (482°F). Li-phosphate enjoys similar and better temperature stabilities than manganese.


Undercharging

Undercharging (4.1 - 0.15 = 3.95 V) results in underusage of capacity - you lose about 20% - 30% of the potential capacity.


  The main challenge in charging a Li-Ion battery is to realize the battery's full capacity without overcharging it, which could result in catastrophic failure. There is little room for error, only ±1%. Overcharging by more than +1% could result in battery failure, but undercharging by more than 1% results in reduced capacity. For example, undercharging a Li-Ion battery by only 100 mV (-2.4% for a 4.2-V Li-Ion cell) results in about a 10% loss in capacity. Since the room for error is so small, high accuracy is required of the charging-control circuitry.


From Analog Devices



From Powerstream