We will begin with a bit of background. Two sets of berachot are recited under the chupa: birkat eirusin and birkot nisuin/birkot chatanim. We refer to the latter as the sheva berachot (which are actually six special berachot in addition to the beracha over the cup of wine that accompanies them). The birkat eirusin relates to the betrothal,[2] and the sheva berachot relate to the nisuin and are repeated through the week of celebration.[3]
Classical sources seem to indicate that fundamentally, the chatan himself should recite at least the birkat eirusin before fulfilling his mitzva of getting married.[4] However, due to one or more of the following concerns, a strong minhag has developed that other people recite both sets of berachot (although some concerns may apply more to one than to the other).
The Mordechai[5] says that it is haughty (yohara) for the chatan to claim the berachot for himself. The Rambam is reported to have said[6] that several of the sheva berachot were instituted to bless the chatan and kalla, and they therefore should be recited by others. Orchot Chayim[7] asserts that we are concerned that if the custom is established for chatanim to recite the berachot, those who are unable to do so will be embarrassed. The consensus is that the chatan should recite the berachot only if he is the sole available person who can recite them reasonably. [8]
There may be a practical difference between the reasons. Some of the sheva berachot are general praises of HaShem and do not focus specifically on the chatan and the kalla. In theory, according to the Rambam’s reason, the chatan should be able to recite those berachot. It seems that in former times, one person recited all of the berachot, and in an “all or nothing” situation, we prefer that the chatan do nothing. Nowadays, when we assign the berachot to a number of individuals, one could contend that the chatan could say the first few sheva berachot. Nevertheless, the prevalent minhag is that the chatan does not recite any of the sheva berachot. This is the correct practice according to the other reasons and, in general, is just as well (the chatan has enough limelight). This is true both under the chupa and during the week of sheva berachot celebrations.
The matter is less clear when it comes to leading the bentching/zimun. On the one hand, leading the bentching is fundamentally the same at a sheva berachot celebration as at other meals, and there should thus be no reason to exclude the chatan. On the other hand, there are clear connections between the bentching and the sheva berachot. The sheva berachot are recited specifically at the end of the Birkat HaMazon; the mezamen[9] waits until the sheva berachot are finished to make the beracha and drink the wine; and the mezamen recites special sheva berachot insertions of “D’vai haser …” and “… shehasimcha bim’ono.” Therefore, it would not be too surprising if the chatan’s ineligibility to recite sheva berachot would extend to being mezamen.
Let us analyze whether the aforementioned reasons to exclude a chatan apply also to being the mezamen. The bentching and the special insertions recited by the mezamen do not focus on the chatan’s and kalla’s welfare. It is unclear whether there is a specific concern that chatanim would feel pressured to lead the bentching and/or be embarrassed if they do so improperly. Finally, it need not be perceived as haughty for the chatan to lead the zimun, considering that this is a normal task and that he is usually a guest of the sponsors of the festivities and not seizing the honors himself. Therefore, one can easily make the case that a chatan can lead the bentching, and some poskim even cite a minhag that the chatan is specifically given this honor,[10] especially at seuda shlishit.[11]
However, since the clearly prevalent practice is that chatanim do not bentch at their own sheva berachot (even if this practice may have been originally motivated by ignorance), it would be objectionable for one to do so without an exceptional reason. Granted, it is unclear whether the practice has reached the level of a binding minhag. Nevertheless, due to the practice, a chatan’s leading of the bentching is likely to raise eyebrows and create an appearance of haughtiness, at least on the level of the spirit of the law. This is all the more so true if it is perceived that he is doing so because he is the most learned person there, as you inquired. However, as we noted above, the chatan may recite any of the sheva berachot if no one else is capable, and we can extend this to zimum when no one else feels comfortable leading the somewhat more complicated bentching.