I am looking for an intuitive explanation for why divergent summations (that are always increasing) have finite values assigned as negative. An example that is beyond "because the math says so" kind of answer.
As examples, the following summations are always increasing and divergent:
$$1+2+3+\dots n+\dots=-\frac1{12}$$
$$1+2+4+\dots2^n+\dots=-1$$
$$\sum_{n=0}^{\infty}a^n=\frac1{1-a}$$
If you input $a=2$, you get the result I predicted. If you input any $a$ such that $|a|<1$, the summation makes sense, logically, but it doesn't everywhere else.
As for the first one, there are multiple places its been seen (Big Bang Theory, the TV show, for example) and by googling $1+2+3+4+5+6+\dots=-1/12$, you should get results as too why.
And as you will notice, they are negative. However, I look for the answer to the question "why they are negative according to multiple agreeing summation methods, which are considered good or strong methods".
The fact that the few examples you've seen come out as negative is just a coincidence, not a general fact. Any "reasonable" summability method is linear (if the method gives values to $\sum_n A_n$ and $\sum_n B_n$, then it gives $\sum_n (A_n + B_n)$ the value $\sum_n A_n + \sum_n B_n$) and regular (it agrees with ordinary summation for convergent series). Take any series with positive terms that your method gives a negative value, add positive numbers to finitely many terms, and you'll get a series to which the method gives a positive value.