New replacement battery charges very slowly

I will start from far away.

In January, I put a new battery into my laptop, a few weeks ago it started to act strangely, at first it could only charge to 85%, then it would stop charging, saying power adapter in but not charging. And it seemed like it was degrading every day, to 75% and even lower, so I thought it is a battery and I switched it. (p.s., it wouldn't even charge if left untouched for the whole night). However, it would charge from 0 to 75% quite quickly, like it normally used to.

Once I replaced it with a new battery, it charges very slowly. After the replacement, I have tried to charge it, no luck. I have tried to do the battery calibration, first draining it and then charging it to 100%. However, after I've done this, it still drains faster than it charges.

Coconut battery also acts strange. Sometimes it shows that battery is charging with 15-30 watts (for a second), then all the time it shows that it is charging with 0 watts.

Also, as it was charging the last night, for about 8 hours I have noticed a strange tendency:

Once the battery charges to 50%, it charges very quickly to a 100%.

So, what I have tried:

1) Changing the data flex cable that connects battery to the motherboard from the one that came with the battery to the one I had with my original battery.

2) Reset SMC and NVRAM

3) Try different charger and a cable. (the original cable and charging brick charge the other Mac fine).

P.S. there are some other strange symptoms:

1) When the battery is charging, the Mac might go to sleep suddenly, and when you click the power button it returns you right where you've been, without asking for a touch id or a password.

2) The battery can be seen as charging in the top menu, though when you are not logged into OS, it does not show as charging, the same with battery symbol on the black background (like when some kind of BIOS starts up I suppose?)

So, my question would be following, is there any chance, that the replacement battery I got is faulty based on the symptoms I have, including slow charging? Because it seems like, even though the power charges can give up to 96 watts, the Macbook does not ask for it at all.

I would rather prefer it is the faulty battery and not something wrong with the power circuit on the logic board, so I came to ask, whether it is reasonable to ask for battery replacement?

EDIT:

Another strange thing I noticed, is that when the top menu says that the battery is not charging and power source is power adapter, it is the only scenario, when the wattage of charge can jump up to 50-60 watts, or at least stay at 20 watts without going to 0: