To start, the idea of charging in parallel while discharging in series is indeed valid. And for multicell battery packs such as for electric automobiles and ebikes, it's the only practical result. That said, the idea can sometimes vary, with some solutions providing the bulk of charging current through the series connection and then having per-cell leads to balance each cell.
In your case, you would have a substantial number of cells in series, to the point that series charging would require high voltage DC, beyond the normal 50-60 VDC that constitutes low-voltage.
But depending on if charging and discharge are mutually exclusive operations, one option would be to electrically break the pack into smaller groups, so that existing charge controllers can charge each group through normal means (ie balancing wires). Supposing that you used 12s charger ICs, that would reduce the number of ICs to about 9 for a pack with a nominal series voltage ~400vdc. You would have to make sure these ICs are isolated once the groups are reconstituted into the full series arrangement.
Alternatively, you could float all the charging ICs, by having 9 rails of DC voltage to supply each of the charging ICs. And this would allow continuous charging and battery monitoring during discharge. Even with the associated circuitry to provide these floating rails, the part count is still lower than having each cell managed by individual chargers and MOSFETs.
It's not clear from your post what capacity or current you intend for this overall pack, but even in small packs, I cannot possibly advise using anything but a proper li-ion charge controller for managing battery cells. The idea of charging a capacitor to 4.2v and then blindly dumping voltage into a cell is fraught with issues, such as lacking actual cell temperature monitoring or even just charging the cell in a healthy manner. Charge IC are designed specifically designed for the task, and are just plain easier to build into a pack while being safer.