It seems standard for a lithium-ion charger to cut off the applied voltage when the CV-mode current draw dips below 0.1C (or thereabouts). Why is this necessary? Why can’t the charger continue to apply 4.2V indefinitely?
Applying 4.2V indefinitely to a LiIon cell will damage or destroy it.
Worst case metallic Lithium may "plate out" and the classic "vent with flame" failure mode may occur.
A LiIon cell is mechanically stressed by charge/discharge cycles due to the transport of Lithium ions in and out of the cell structure. [LiFePO4 cells almost eliminate this affect by providing a mechanical olivine "cage" into which the Li ions are "intercalated".
A look at the charge/voltage curve for a charging LiIon cell shows that towards the end of charge the voltage rises at a rate which is increasingly greater than the rate of charge absorption. This reflects an increasingly inefficient charging process as the last available areas are used to 'store' ionic Lithium.
Bythe time Ich under CV mode has dropped to 10% of Imax the cell is almost completely charged to capacity - "road warrior" mode where maximum possible energy is stored at the expense of lifetime. Cells charge terminates at 0.1 x Icv will have a short cycle life. Terminating at 0.5 x Icv still achieves > 90% of maximum capacity and stopping at 0.25 x Icv is probably preferable to 0.1 x Icv in most cases.
Proper design of charging circuitry makes it easy to detect current into and out of the battery while load varies from 0 to 100%. The battery and charging circuitry are fed from the charger input point as is the load. The load current may affect the amount of current available for charging (depending on charger capacity) but battery current is monitored regardless of direction or magnitude.
When charging multiple cells in series a modern BMS monitors the voltages of ALL cells individually. When one or more cells in a string reach their final state of charge ahead of others balancing circuitry 'shunts' current around the cell so that the cell is effectively separated from the charging string.
All lithium-ion cells will continue increasing in voltage until the charge current is cut. So unless you have a damaged cell, the cell's voltage will pass 4.2 and can go as high as 4.5 or 4.6 V. So it is possible that with an ideal CV supply, to keep it at 4.2 V, and let the cell naturally cut off the current as its voltage rises. A cell receiving above 4.2 V will be incrementally damaged.
The problem arises when the CV supply is not perfect. I.e. as the current diminishes, its voltage would rise and exceed 4.2 V. Or a malfunction with the charger will make it supply 4.6 V instead of 4.2 V, at very low currents. So lithium-ion chargers in order to account for a malfunction of the supply, would want to cut the current at some stage. Some chargers have a timer, and all chargers will cut the current if it drops below a threshold (0.1C or even lower).
Then there is the problem of damaged cells that refuse to go above 4.2 V. In this case there should be the timer or undercurrent detector.
Also note that balancer-protection boards usually cut off above 4.2 V. In a series configuration, the balancer-protection board will cut off when any one of the cells exceeds the cut off voltage. It can be as high as 4.35 V given the tolerances, and this can damage the cell. That is why you should not rely on the balancer-protection to keep the cell voltage below 4.2. In a nS series connection, I recommend the charger to cut off at n * 4.2 V, and not rely on the balancer protection to kick in. Note that not all balancers have protection circuitry and thus may not cut off the current. Also note that balancers shunt the cell with only a minor current, as little as 50 mA. Thus at high-C charging, balancing may not happen if the cells in series are significantly different. I also believe that after cut off, whether by the charger or the balancer, balancing will continue until each individual cell drops below the balancing voltage (which could be as high as 4.35 V).