See https://www.dwds.de/wb/ur- which is very helpful, especially when you have a look on the etymology. You are right in stating that ur- is used to refer to the origin of something far back:
- bezeichnet den Anfang, das erste, etw. sehr weit Zurückliegendes"
other examples, but with interesting semantic nuances (when you take care of the literal meaning of the nouns used in the composita) are
- Ursache (cause)
- Ursprung (origin)
The second meaning
- bezeichnet Echtheit, Unverbrauchtheit, den idealen Zustand nach der Entstehung, Produktion von etw.
genuineness, authenticity is raising from the first. To my eye the most interesting example here is
which combines the semantics of authenticity and origin:
An urtext edition of a work of classical music is a printed version intended to reproduce the original intention of the composer as exactly as possible, without any added or changed material."
Notabene that the english genuine (with the etymology of latin gignere "beget" - see https://www.merriam-webster.com/dictionary/genuine) - also relies on the semantic connection of origin and pureness, authenticity Also take notice that origin has strong etymological links to ur-, see https://www.dwds.de/wb/original - the or- in origin correlates to ur-.
The third meaning
bezeichnet einen hohen Grad
a high degree is the explanation for your examples
- urplötzlich (Caution: you wrote "ursplötzlich, which is wrong) and
- urschön (which I have never heard before, to me it sounds like some regionalism)
In youth language there exists (or at least: existed, when I was youth) the superlative urst which is used to denote a very high degree of something ("urst schön", "urst langweilig"). I think that the grammatical interesting construction of a "superlative" of a prefix (which does not exist in standard german!) is a typical means of youth language to demonstrate deviance of the norms of the adults.
Related concepts in german language
Caution, the following is somewhat speculative and must be read cum grano salis:
In german language there is a semantic connection between concepts of authenticity, pureness and origin. As a very prominent example for the first take the title and concept of Immanuel Kant's "Kritik der Reinen Vernunft" where "rein" ("pure") means "unvermischt" (unmixed) but also clean, neat)
Ideas which are blending concepts of pureness, origin and authenticity are raising especially in german Romanticism (consider for instance "Die Christenheit oder Europa" by Novalis), but have german predecessors - take, for example, Martin Luther with his concentration on the bible as the pure origin of Christianity. (But this is very associative and maybe loses the track of your question.)
That's why I was not surprised when I learned that english language uses a germanism to denote Urtext.