As we would expect of a debate between learned rabbis, each side has a significant basis. Our approach incorporates the claims of each side, hopefully in a balanced manner.
One related case with classical sources on the topic of activity before davening concerns haircutting, which is halachically equivalent in many respects to cutting a beard. The gemara forbids haircutting before Mincha lest one become so occupied with it that he ends up not davening.[3] The Shulchan Aruch[4] says that no formal prohibition was made on haircutting before Shacharit because it is an uncommon occurrence. However, the Eliya Rabba[5] maintains that the Shulchan Aruch refers to the time before alot hashachar (at least 72 minutes before sunrise). After that point, when davening is nominally possible, it is forbidden to take a haircut, because all work is improper before davening. Shacharit is more lenient than Mincha only regarding the added stringency of not starting haircutting a half hour before the time of davening, but not concerning the regular regulations on activities. Although the Eliya Rabba cites the Kolbo as saying that haircutting is not the type of activity that is considered forbidden work, the Mishna Berura[6] accepts the strict view. Thus, the first opinion you cited certainly has validity.
It is important to consider, though, that the classical sources dealt with haircutting, which is similar to shaving only for those with beards. That was indeed the norm for religious Jewish men until the modern era. For such men, shaving is a periodic activity that we view as a matter of choice at any given time. For many clean-shaven men, on the other hand, daily shaving is a matter of simple hygiene that cannot be delayed for long. This distinction can be significant in several areas of Halacha, including shaving on Chol Ham’oed, as well as during sefirat ha’omer and the Three Weeks. For example, those who permit shaving on Chol Hamo’ed, based on the lenient rulings of Rav Soloveitchik and Rav Moshe Feinstein,[7] reason that shaving before Yom Tov does not suffice for an entire week, as it once did. There are even (disputed or contrary) accounts that Rav Feinstein maintained that not only may a clean-shaven man shave on Chol Hamo’ed, but it is preferable to do so in honor of the holiday.
Some apply the same logic to shaving in the morning. The halachic concept of hikon likrat Elokecha mandates preparing oneself with a clean body and appropriate clothes for davening.[8] One posek who has written that shaving may be included in the category of hikon likrat Elokecha is Rav Yitzchak Yosef,[9] although he describes shaving before tefilla as a legitimate but not preferable practice. According to many oral accounts of rabbinic advice, especially in societies in which “needing a shave” has a social stigma and is a physical nuisance, shaving before Shacharit is even preferred, and this is the practice of many fine Jews.
That being said, it is unwise to make a single, sweeping rule. Those with beards should normally not shave before Shacharit. Those with slow growing or light beards would do better to shave before going to sleep or after davening, and if this is not satisfactory, they should wait until after having said Shema and some berachot.[10] Those who need a morning shave to make themselves presentable should feel free to do so before davening. However, if they are running late, it is hard to justify missing even P’sukei D’Zimra on that account.