To summarize the question:
We can say the first trial stemming from as well as the first
trial to stem from. So why can't we say the first person
climbing Mount Everest as well as the first person to climb Mount Everest?
In Swan's examples each verb (win, climb, arrive) is what Quirk et al. in A Comprehensive Grammar Of The English Language p201 (henceforth just Quirk) would classify, according to a complex verb tree, as: Dynmamic > Durative > Conclusive > Agentive > Accomplishment.
One of the verbs that Quirk lists as falling into the above category is write. An example of write as a conclusive/accomplishment (using one of Swan's the oldest/first/only, etc constructions) is:
- Matt was the only child to write a 'thank you' letter (i.e. conclude or accomplish the action of writing it).
Swan's sentence Who was the first person to climb Everest without oxygen? will most likely be similarly interpreted as the accomplishment of getting to the top of the mountain.
These two sentences can be expanded to use the past simple in a relative clause:
- Matt was the only child who wrote a 'thank you' letter.
- Who was the first person who climbed Everest without oxygen?
But Quirk also lists write in another category, namely: Dynamic > Durative > Nonconclusive > Agentive > Activities.
An example of write in this category is:
- Matt was the only child writing at the end of the test (i.e. the focus is on the nonconclusive nature of an action in progress).
Similarly, it is possible to imagine a context using Swan's the first person + climb example where the focus is on the activity rather than the accomplishment. For example, describing an obstacle race:
- Mae was the first person climbing the wall, but Ximena was first over it.
These two sentences can be expanded as:
- Matt was the only child who was writing at the end of the test.
- Mae was the first person who was climbing the wall, but Ximena was first over it.
Another verb that falls into both the conclusive/accomplishment and the nonconclusive activity categories is read:
- Emiko was the first student to read Great Expectations (i.e. accomplish the task, reading it to the end).
- Emiko was the first student reading at the beginning of class (i.e. to get out her book and start reading).
So, in summary, the answer to the question why is it that you can't use '-ing' instead of 'to infinitive' in the following sentences? is:
You can, for climb and other verbs which fall into both the two categories listed by Quirk, where the focus is respectively on the conclusive/accomplishment nature of the verb in context or on its nonconclusive/activity nature.
The OP states of the two trial texts that use the -ing form and to-infinitive form of stem respectively: 'both versions seem to be well-formed'. This seems to be the case for me too. However, it raises an interesting issue.
If we expand the -ing form in This is the first trial stemming from the investigation of Robert S. Mueller in a similar way as above, we get:
*This is the first trial that is stemming from the investigation of Robert S. Mueller.
But stem in this sense of the word is a non-progressive verb, so the expansion results in an ungrammaticality. In other words, This is the first trial stemming from the investigation cannot be regarded as a reduction in the same way as the first person climbing is a reduction from the person person who is/was climbing.
There is a discussion of this phenomenon in the chapter on Relative Clauses in The Grammar Book: An ESL / EFL Teacher's Course (p597):
Participles that do not derive from relative clauses
Our discussion of this topic is incomplete unless we point out that
there are also adjectival -ing participles that could not possibly
be accounted for via the pronoun + be deletion rule. Recall that
verbs which are stative rather than dynamic rarely take the
progressive aspect, yet stative verbs can and do occur as adjectival
- Max built an additional room, measuring 12 by 12 feet.
This -ing particple cannot be derived from a reduced relative
- *Max built an additional room, which is measuring 12 by 12 feet.
How to account for the structure of such adjectival participles
remains an unsolved problem in need of further research.
Note: that The Cambridge Grammar Of The English Language (CGEL, p446) does not use the term reduced relative clause for -ing constructions such as above. They are classified under Post head modifiers > non-finite gerund-participial clauses. Example: People living near the site will be seriously disadvantaged.
And one last point, the CGEL (p118) has a simplified version of the Quirk's verb classification tree mentioned above, but includes the terms telic and atelic as follows: _ Accomplishments [telic] and Activities [atelic]. There is a discussion of these terms on ThoughtCo: telicity (verbs).