Is Schmach not 'disgrace' rather than 'taste', which would be 'Geschmack'?
First off: yes, you are - somewhat - right. But ....
Translating from one language to another is not only replacing words and changing the sentence structure. Quite often it is also a projection from one cultural background to another. A word has not only a meaning, but is also a representative of a certain way of thinking about reality, a concept. e.g. there is no genuine german word for "fairness" and there is not genuine english word for "Fleiß". Yes, both can be translated literally, but none of those translations would really convey the meaning the respective words have in their original language.
Having said this, let us come back to the example at hand:
des Leidens herbe Schmach
"Schmach" might in some cases be translated with "disgrace", but here the better translation would be "humiliation", IMHO. To better explain what "Schmach" means, here is another translation: the famous monologue of Hamlet, in the translation of August Wilhelm von Schlegel:
The insolence of office and the spurns
That patient merit of the unworthy takes,
Den Übermut der Ämter und die Schmach,
Die Unwert schweigendem Verdienst erweist,
You may, as an aside, notice that "patient merit" is translated "schweigendem Verdienst", not "geduldigem Verdienst" - for the same reasons I explained above. You may also notice that "patient", "patience" comes from the latin root "patientia", the willingness/ability to suffer or endure (misfortune) and to suffer loudmouthed idiots without saying anything is a quite fitting picture of "patientia", don't you think?
Now, have a look at the (german) word "herb". "Herb" is an Adjektiv and can be translated with "dry" or "tart" (for wines), "austere", "bitter", "harsh", "acetous" and so on. You will notice that all these translations circle around taste impressions. This is the original meaning of the word: a certain taste or class of tastes. Yes, the word "herb" can be (and is) used figuratively as well and sometimes the figurative and direct meaning meet in the translation, like in:
die herbe Schmach der Niederlage
the bitter taste of defeat
a phrase you might find in a newspaper after your famous football team lost against a perceivedly abysmal opponent.
Putting that all together: Jesus was (according to christian hagiography) tried unlawfully and sentenced to death on phony charges. So, his treatment could be called "herb" (here the figurative meaning) and because he was not just suffering, but unjustifiedly so his suffering is not called simply "suffering" ("Leiden"), but called "Schmach", in a similar meaning as in the Shakespeare quote.
The speaker does the same, claiming that, because Jesus did it originally and he follows him, all will go well in the end. Here the "herb" and the "versüßen" (to sweeten) create an opposite, a common figure of speech. It was bitter before but in the end it will be sweet.