The ערוך השלחן in 202:12 writes the following:
עליהן ברכה מבוררת יותר מפני חשיבותן שכל
החשוב יותר ברכתו מבוררת יותר וליתן שבח
להקב״ה שברא דבר חשוב כזה
[The Rabbis] established a more precise blessing on them because of their significance, as everything that is more significant has a more precise blessing to give praise to Hakadosh Baruch Hu that he created such a significant thing as this.
According to the ערוך השלחן, a more specific blessing has the accomplishment of praising Hashem for a more significant or special creation. By making the more specialized blessing we are being "stringent" on making sure to properly praise Hashem for the creation of this food.
Therefore, the hierarchy would be as follows:
1. The prohibition of G-d's name in vain. Avoid a blessing in vain - if there is doubt as to the validity of a blessing, do not say it.
2. The praise that comes from the blessing. Say as accurate a blessing as possible in order to express the maximum level of praise. If one can justify saying a more precise, higher praise blessing by taking another food item which undoubtedly deserves that blessing, do so.
3. Saying a correct blessing. If there is a question as to which blessing is appropriate for this food, say the more inclusive one. (This level would essentially always be covered by #1, but still exists as per the Gemara that prohibits enjoying this world without making a blessing)
In a nutshell, ספק ברכות להקל would mean being lenient on the praise to Hashem that comes from the blessing due to the concern of the prohibition of taking G-d's name in vain.
רבינו יונה in his commentary to the רי"ף in Berachos, 28b in the pages of the רי"ף, in discussing why one makes a בורא פרי האדמה before a שהכל, says that the former is מעולה, higher level, because it is פרטית, specific. It seems that the advantage of a more specific blessing is that itself - that it is more specific. Perhaps this is similar to the requirement of רבי יהודה, because of ברוך ה' יום יום (See Berachos 40a) that we have a responsibility to give Hashem specific praise (as opposed to higher level praise). This being the case, the "stringency" of making the accurate blessing is being stringent on the requirement to make a precise blessing.
The ט"ז himself actually makes this point very clearly, in Siman 202 Sif Katan 4 - in discussing the blessing on jam and pulverized fruit, he explains that even though the Terumas Hadeshen says to make a Shehakol, one can a priori make a Ha'etz since (as he concludes) it is the "main" opinion, and יש לה מעלה על ברכת שהכל כיון שהיא ברכה מבוררת - it has an advantage over Shehakol since it is a more select blessing.
(It should be noted that this is not a contradiction to the ט"ז cited in the question, who said to bring a different fruit which is definitively Ha'etz, as opposed to here where he recommends to even make a Ha'etz on the food item in question itself, as in 202:13 he was discussing a situation in which there was one preferred opinion among many existing opinions, whereas in this case he is suggesting the possibility that there really is no conflicting opinion.)
R' Sheinberg ZT"L in משמרת חיים חלק א שאלה ח writes a similar thing, that the reason Chazal made a beracha of Ha'etz or Ha'adama is because of the need to make a specific blessing, i.e. that specificity is in itself a value. (His idea is difficult, because he is using this approach to answer a question on the Mishna Berura, but the Be'ur Halacha writes explicitly in the beginning of Siman 202 that the blessing of Ha'etz is because of its חשיבות, significance, like the Aruch Hashulchan above.)