I clamp the 741 with a reference voltage of ~13.7V (using 9V+4.7V Zeners) and expect charging to stop there, signalled by the LED turning off. In reality the battery seems to just stay there but the LED never goes off. If I pull a the 24V into the 741 input instead, I DO see it turning off. I suspect there is some feedback here which I don't quite follow.
Feeding +24V into the 741 could damage it, and will at least disturb its normal operation, because it is powered from the output of the LM317 regulator, at about 17.2V. You're not supposed to force any pin of any IC beyond the voltages on its power supply pins except in certain cases like open collector and open drain outputs where it's specifically allowed. Many ICs will be permanently damaged if you do this.
Do I need to have a diode between the TIP122 emitter and battery to prevent current in reverse direction (when the 24V main source is off)?
Yes. And when you add it, take your negative feedback (to pin 2 of the 741) from after the diode, i.e. from the battery (actually, through a resistor), so the 741 regulates the output voltage at the battery, not the voltage before the diode.
In my tests I never saw more than 0.2A going to the battery, but I am not sure why. Is this a result of the TIP122 limiting current (i.e. base current * hFE)?
Not if you're using a TIP122. A TIP122 is a Darlington transistor, and should be drawn using a Darlington transistor symbol on your schematic. The TIP122 has a minimum current gain of 1000, and the 741 can supply at least 10 mA, so the TIP122 should be able to provide at least its rated current of 5A.
My guess is that the battery is already charged to roughly the target voltage, and won't draw much current unless you increase the charge voltage - i.e. increase the zener voltage.
General comments on your design:
There is no charge current limiting. The configuration you have, with a reference voltage feeding an op-amp driving a Darlington transistor, is a voltage regulator. The op-amp will control the transistor so that the output voltage (at the transistor's emitter) is equal to the reference (zener) voltage. If the battery's terminal voltage is lower than the reference voltage, the regulator will still try to apply the reference voltage to the battery; this may cause the battery to draw a lot of current. You need some kind of current limiting.
I don't see any need for the LM317 regulator. You can just power the whole circuit from 24V.
Zener diodes are inaccurate and their voltage varies with temperature. The target voltage for the battery should probably also be adjusted according to temperature - all battery chemistries are affected by temperature - but the chances that the zener voltage variation will match the optimal voltage variation for the battery are not very high!
You need protection against back-feeding from the battery when the input supply is not present. A diode in series with the emitter will isolate the transistor, and a resistor from the battery terminal to the inverting input (pin 2) of the op-amp will prevent damage to the op-amp when the input supply is not present.
The LED won't go out unless the battery is charged to a slightly higher voltage than the reference voltage. Without that condition, the 741 will need to keep the TIP122 biased ON to pull the battery terminal voltage up to the reference voltage, and this will illuminate the LED.
Your transistor will potentially have to dissipate a lot of power. You could avoid this by using a more efficient charger based on a buck switching converter.
There are ICs that will do all of this, and the other features that you mentioned. Their manufacturers have sample designs in the data sheets, and they provide very helpful application notes. Have a look on digikey.com in the Integrated Circuits category, under various headings such as:
PMIC - Battery Management
PMIC - Current Regulation/Management
PMIC - OR Controllers, Ideal Diodes
PMIC - Power Management - Specialized (perhaps)
PMIC - Power Supply Controllers, Monitors
PMIC - Voltage Reference
PMIC - Voltage Regulators - DC DC Switching Controllers
PMIC - Voltage Regulators - DC DC Switching Regulators
PMIC - Voltage Regulators - Linear (LDO)
PMIC - Voltage Regulators - Linear + Switching
PMIC - Voltage Regulators - Linear Transistor Driver
PMIC - Voltage Regulators - Special Purpose