How does 儘 + 管 compound to mean "despite”?
One of the meanings of 儘 is no matter (任凭,纵使),

《左传·文十四年》：公子商人，尽其家贷於公。
《礼·曲礼》：虚坐尽后，食坐尽前。俗作尽。

So 儘 + 管 could be translated (literally) to no matter what the influence is


I'm just a native speaker, not a profession. I don't have an answer here, but I may share some of my thoughts that might be helpful.

In the second image, do notice it is 盡 not 儘 (however both characters simplify to 尽, making 尽 has 2 pronunciations(jǐn and jìn)
And in 儘管 it (only) pronounces jǐn
It shall be notice that compounding Chinese characters to words doesn't simplify compound their meanings.
Traditionally, I mean in Classical Chinese, 儘 already means despite. However the meaning of it twisted slightly (in most cases it means to the greatest extent  and always)
The same thing goes with 雖然 and 但是. If you split the characters and find out each of their meanings, you would find that 雖 and 但 literally mean the same.