I'm not sure I'd call the interpretation you quote "programmatic" exactly. In fact, this sort of characterization might in fact be sort of the essence of absolute music.
As you likely know if you've been reading about this, Brahms was a friend of Hanslick -- the music critic and sort-of philosopher -- who famously characterized music as "tonally moving forms" or "sonically moving forms" or something like that. (The original sentence is notoriously difficult to translate in its subtleties: "Der Inhalt der Musik sind tönend bewegte Formen.")
Something inherent in that description is a sense of motion and temporal development. Music is not a static entity to be studied only on the page for Hanslick. But music's meaning and effects don't come from extramusical programmatic ideas -- they are inherent in the sounds and language and relationships within music itself.
Brahms had slightly different things to say about music, but he shared something of this view of absolute music as being built on a kind of intrinsic musical logic that taps into a deeper meaning than any "program" or series of Wagnerian Leitmotifs or whatever could express.
Whether Brahms himself would use the evocative term "war" between major and minor, I don't know. But the first symphony certainly has a dramatic arc to it, created partly by the conflict in keys (among other things).
Absolute music doesn't lack drama. The temporal nature of music requires drama, in fact. According to 19th-century German aestheticians, it just might lack extramusical references to create that drama. As Hanslick writes:
The most important factor in the mental process which accompanies the
act of listening to music, and which converts it into a source of
pleasure, is frequently overlooked. We here refer to the intellectual
satisfaction which the listener derives from continually following and
anticipating the composer’s intentions – now, to see his expectations
fulfilled, and now, to find himself agreeably mistaken. It is a matter
of course that this intellectual flux and reflux, this perpetual
giving and receiving takes place unconsciously, and with the rapidity
of lightning flashes.
The act of listening is therefore about hearing the give and take, the satisfaction of following a musical pattern to its fulfillment, as well as the surprise when it heads in a different direction. It is still a drama, perhaps one might even say a type of narrative, just one that takes place among inherently musical ideas (rather than representing other things).
We don't need to fall back on some story about alphorns on a mountaintop heralding the coming of a great hero to characterize what's going on, for example, in the first Brahms passage you linked to. We could instead talk about the inherent tonal tensions, the role of dissonances and clashing harmonies, and perhaps the ultimate arrival of major to resolve the earlier tensions of the piece.
That latter type of interpretation is close to what folks like Brahms and Hanslick saw as the true aesthetic way to appreciate absolute music.
It's when you start talking about "symbolic redemption, transformation, or victory" that things get a bit more dicey. In the Hanslickian view of the world, it's probably okay to discuss a "transformation" from minor to major, perhaps even a type of conflict between them and maybe even a metaphorical "victory" of one over the other. All of this is just seen through the lens of the composer and the sounding music conveying musical patterns that are reinforced, then with expectations denied, then finally with a new path emerging tonally at the conclusion. But if you dive deeper into the "war" and "victory" metaphors or begin to speak of "redemption," you're imputing human emotions or personifying the music in a way that is not necessarily about the musical patterns exclusively anymore. It's now a psychological reading, to which Hanslick would retort -- "whose psychology"? The composer's? Yours? Some fictional narrative figure or "hero" of the piece?
I'm not arguing in favor of one side or another, by the way. But the absolutists never denied an element of a temporal arc within music that can create drama and emotion for listeners. It's just that they preferred to say all of that came from the logic of the music itself (with its melodic and harmonic twists and turns) rather than depending on some external program.
To use an analogy, consider a painting, say a landscape. A viewer might appreciate technical subtleties like different types of brushstrokes or the use of color palettes or color relationships, but most viewers are also concerned with how the painting evokes an actual image or scene (and how well it does so). Now imagine an abstract painting, with no discernible subject or attempt at representation. At this point, discussion of the aesthetic qualities often becomes solely about technique -- contrasts in color, effects of brushstrokes and the like. I suppose in art criticism, there are some extreme formalists who would say to forget about the representation in painting as well, and to focus on the technical, even in landscapes.
For those who adhered (and still adhere?) to the absolutist philosophy about music, they would say music should be appreciated in that manner of abstract painting. Any attempts at "images" or "evocations" of non-musical meaning are poor representations at best. So we should try to explain music's impact by focusing on the technical -- the contrasts of color, the sweeping brushstrokes, etc. A contrast or conflict with major and minor as a structural device falls under that category, to my mind.