The explanation that you are giving for the difference between "aber" and "sondern" sounds very good to me as a native German speaker. What I can add is:
- „nicht [a], sondern [b]”: means [a] is not the case, but [b] is the case. Contrary to [a]. (a sidenote: as a German speaker I find it weird somehow when other languages use the word meaning "aber" here). Other words in English may be "instead" or "rather".
- „nicht [a], aber [b]”: this is basically the word "but" in all its more complicated or nuanced meanings: to express reservation, to correct, to add additional information or something that is against the listeners' expectation. So it means [a] may not be the case, but still [b] is the case. Or: Just because [a] is the case, it doesn't mean [b] isn't the case. English synonyms could be "yet", "nevertheless", or "but still".
If I had to translate the English sentence "It's not an exponential increase, but a linear one" into German, I would surely translate it with "sondern": "Es ist kein exponentieller Anstieg, sondern ein linearer". This would emphasize that the grow is linear = not a big increase. In the context of the Corona pandemic, that would imply that he would mean we do not need to take stricter measures, because the increase is not so bad.
As far as I understand him, the reason why the Austrian Health minister used the word "aber" here is to emphasize that there is an increase: "It's not an exponential increase. But still, it is an increase". This emphasizes not the adjective, but the noun ("increase"). In the context, this implies that he means we must take stricter measures, because an increase is still there.
Stylistically, it would have been good if he had added the word zwar: Es ist zwar kein exponentieller Anstieg, aber ein linearer.