How can I erase "Biological Treatment" in this example image, which is attached. Note the background is a 2D gradient, so what I could use is, for example, a tool that lets me select a rectangular area around the text to be erased, and tell it to "erase to context image gradient" or some such.

There used to be an "interpolation plugin" that did something like this for images, but it doesn't compile under GIMP 2.8, as exemplified in the image of the duck attached as a file to this message. In that example, the patches on the duck pond are selected areas that are erased by interpolating the gradients surrounding those patches.

It's a png so layers data are lost.
The best way i think would be to isolate the tank and arrows then redo the gradient yourself.

I think you are talking about the "resynthesizer" plugin which is very common (and is part of Partha's build if you are on Windows, and in the 'gimp-plugin-registry' package if you are on Ubuntu or derivatives).

I remember that duck example and the plugin, alas I can not get it to compile (linux) either.

Probably better ways in any case. Resynthesizer + heal-selection as Ofnuts mentioned or maybe the g'mic plugin. http://www.gmic.eu
example gmic morphological inpaint.
Looks like that is just part of a larger image so you might need some variation, but paint in a mask over the text, be generous, not too tight.

gives this:

Is this the plug-in you wanted compiled?