Now I have two devices - an e-ink book reader and an Android tablet (in the picture) - with broken displays and still decent batteries. Their displays don't seem repairable (mechanically crushed). Is there a way utilise their batteries at least?
The 3.7 V / 4 Ah battery (the tablet) is equivalent to 2x 3-NiMh R10 battery packs and it's much compacter.

Device onboard charging circuit could be used for charging but batteries seem glued firmly to the board and I'm afraid that prying would damage them. There are some resources on the internet but those assume that the battery is already dead so no special care needs to be taken. Does anybody have an experience with their safe detaching? Thanks.

For what you want to use the battery ? Maybe your option is to embed it together with the host device in your application

First be aware the electronic packed with the battery (the pcb stripe) is not the charging controller, but only the battery ptotection. Its function is during charging to prevent fire when the real charging controller fails to limit the cell voltage, the additional function is to cut out the load once the cell is fully discharged. Although it may seem to behave in a similar way as the real charging controller does (cut out the charging once the voltage reaches certain level), using it that way actually means overcharging the cell so degrading it rapidly (the protectors use to be set to about 4.3..4.4V, while charger is supposed to be set to just 4.2V; a significant difference between long reliable service vs death within few weeks). Plus you loose one safety layer, as once the protector mosfet fails, you wont have anything to stop uncontrolled overcharging and fire.

The charging controller is part of the main board, usually part of the power management IC, sopractically impossible to extract and reuse from the main board. But known far east sites carry a lot of charger modules, so I would recommend using those.

Regarding the battery extraction: You should make sure you dont separate the layers of the cell, otherwise it is high chance it will swell and fail.
Using a nylon string to cut through the glue (guide it underneath the cell, then use it as a kind of rope saw) uses to work, if gentle pulling is not enough to free the cell.

For what you want to use the battery ? Maybe your option is to embed it together with the host device in your application

My original idea was using it in a pocket worklight (something like the right one here) which is powered by 3xAAA, thus making it operate ~5 times longer). The non-displaying tablet was intended as a recharger, with a connector soldered to the place of the fixed black/red wire.

First be aware the electronic packed with the battery (the pcb stripe) is not the charging controller, but only the battery ptotection. Its function is during charging to prevent fire when the real charging controller fails to limit the cell voltage, the additional function is to cut out the load once the cell is fully discharged. Although it may seem to behave in a similar way as the real charging controller does (cut out the charging once the voltage reaches certain level), using it that way actually means overcharging the cell so degrading it rapidly (the protectors use to be set to about 4.3..4.4V, while charger is supposed to be set to just 4.2V; a significant difference between long reliable service vs death within few weeks). Plus you loose one safety layer, as once the protector mosfet fails, you wont have anything to stop uncontrolled overcharging and fire.

So if successfully detached (including the pcb stripe) and recharging the battery (now built-in to the worklight) using a cable going to the original black/red wire points on the main board (powered via USB from a 5V adapter) should work. Worth trying.

Thanks guys.

Yes, using the original tablet board as a charger will work, however personally I would see as more practical to buy the charger module (any "18650 charger module" will work fine and cost barely a dollar) and build it inside of the flashlight, so tge complete device will be able to use any USB charger...

Won't the 18650 module also have constant current setting thats suitable for 18650 (and could overload a small tablet lipo) ?

The tablet accepted fast charging from the yellow USB port on an ECR501CU3E front panel, which is actually connected directly to the ATX power supply (no data, fast charging only).
According to specs, its charging current is 2.4 A and the complete charging of the 4Ah battery was less than 2 hours.

While fast charging, the tablet was only mildly warm. On the other hand, the remaining battery capacity dropped faster to 90% (compared to slower charging from an 1A AC adapter) and it also probably shortened battery life.
This McIgIcM 5pcs Micro USB 5V 1A 18650 TP4056 Lithium Battery Charger Module should be probably fine with its 1A charging current.