This is a 820-3787-A. The board does not have noticeable water damage.

With the battery plugged in, I need to keep the button down for 6 seconds to turn it on. If I hold for less, it either does not turn on or the chime start but it is interrupted as soon as I release the power button. I need to release it just after the chime is over.

Assuming I timed correctly, the laptop turns on but it is very slow. Even the animations at boot if I press "Alt" are super slow. When the OS loaded, the kernel_task uses more than 400% CPU.
If it goes to "sleep", it never wakes up.

Fans and temperatures are normal. The battery also charges.

I reinstalled the OS and it did not help.

If I open the laptop and physically unplug the battery and I turn the computer on with the power adaptor, the laptop works perfectly well.

AST reported:
- a generic MLB F:050000 code - I think it was relative to the battery but I could not find anything on Google.
- a failed test: TIM Test 75 percent guardband. - Check that CPU heatsink is properly attached, using product spec theta dh + 75 percent guardband - on Google I did not see anything very actionable. It is a stress test and considering that the temperatures were normal, I would not expect any improvement changing the thermal paste or the heatsink. Am I wrong?

iStat says that the battery is always using 18.2A (Charger BMON - I think it is SMC's register IPBR).
The value is the same when the battery is disconnected. The same when the adaptor is plugged in.
This is supposed to be the current from the battery. 18A cannot be right. I am pretty sure that this is the problem.
I think the SMC slows everything down because it recognized that BMON is off.
R7150, R7151 and R7152 seems fine.

What can I check next?

What about "hack" it.
I could put a 10K and 100K from ground to 5V and connect the middle point to SMC_CHGR_BMON_ISENSE/SMC_ADC7 so the SMC gets 0.5V. What do you think ?

you already check R5423 is 45.3K
also try to change the U7100

R5423 is good. I can try to change the U7100 but I was wondering if I need to check something else.

Use ASD, not AST.

The slowness and high CPU use in OS are caused by the SMC. Usually this is a symptom of SMC failing to read some sensor/s.

I think you should start by troubleshooting the battery charger circuit to find the fault.

Disconnect all power from the motherboard, then put your multimeter leads on pins 17 and 18 of U7100 and check resistance between those pins. Should be about 2.2 Ohms. If it shows anything higher, check R7151, R7152 resistors.

If you did not find anything wrong there, replace U7100.