Interesting question. Several things to consider here.
First of all. You can not print one ink at 200%... Unless you can.
When you send something to be printed, and you have no real control over the print process you can not. The maximum you can send is the 100% capability of the device. This is true to digital print systems or traditional print systems, like sheet-fed offset printing.
This limitation is not only defined by the system itself, for example on a digital print you can not force the nozles to shoot more ink. But sometimes it is defined by cost. For example, it could need an additional pass, a 5 ink process. Some providers can not do that, because your project is not the only one in the line of production.
When you need deeper black, for some reason, you have several options.
The normal options:
Use a rich black. Either a "recipe" black, like c50%k100%, or the black defined by your color profile.
Replace the parts you need with a spot ink, which is less transparent-ish than process black. You could end with a 5 ink print. If the provider is small, he can be more flexible, and you could have, for example, a 1 ink print.
If you have a really good understanding and workflow with your print provider.
Using darker ink, spot ink, not process black. If you are not printing photos, but instead your design is something like a "Line art" poster, but you are still using a full CMYK system, you could ask the process black to be replaced with a spot ink.
If the design allows it and the print provider agrees, YES you can use 2 passes of the exact same black plate. This is easy when the press is only a 1 head machine.
You simply take the pile of paper and feed the machine again, thus, actually printing twice, or 200% of the ink.
I have done it only once in my life, and it was to fix a mistake. I had a photo with a big rectangle with only k100% over it. The problem was that the "overprint" option was turned on when the plates were made.
What overprint does in this case, is printing the C+M+Y components of the photo underneath, and a rectangle of black ink, actually showing the photo under the rectangle.
What we did was printing a second pass of the black ink. Then the rectangle was so intense that covered the phantom photo (that was already printed)
But this was only possible because the machine was only 1 head, the C+M+Y was already printed, therefore all the paper already used, the black was not so intense on the photo, that it would not look too dark and dirty, and the black plate was the last one used, and I had a good relationship with the provider.
But the reason was NOT the ones you are mentioning.
A. If your text is small, you do NOT use rich black, unless the quality of the provider is superb! wich in fact will recommend NOT using rich black.
B. If your k100% small text is at risk of being too gray... is because your provider is not good.
Millions of projects are printed with small texts with K only, magazines, books, etc. People know what they are doing. So they print the text correctly.
One additional thing is that the human eye will not perceive a small text, even if it is really a bit light on the black, like gray, because it is surrounded by white, which by adjacent contrast, makes the black darker. So stop worrying about that.
In real-life practice:
Learn how and when you really need rich black, and when you should not.
How and why using one spot ink.
So discard almost completely the other options I mentioned. Including k200%.