Shraddha can be performed even by a Shudra too. In fact, as per the Smritis, it is one of the only two religious rites, that he is allowed to perform. So, it does not require the performer to be only a Brahmin to be considered qualified.
From the Yajnavalkya Smriti's (YS) Chapter 1:
[He should be](i.e a Shudra should be) devotedly attached to his wife,
be of pure conduct, a protector of servants and given to the
performance of Sraddha. With the recitation of the Mantra "Namas", he
should perform the five Yajnas.
The performance of Shraddha requires reciting the Mantra "Swadha" and it is a Mantra which can be uttered by everyone, even by the uninitiates as the following verse shows:
Manu Smriti 2.172. (He who has not been initiated) should not
pronounce (any) Vedic text excepting (those required for) the
performance of funeral rites, since he is on a level with a Sudra
before his birth from the Veda.
So, anyone, who knows how to do it correctly, can perform Shraddah on their own. There's no prohibition on that. But, it is better to appoint a Brahmin, in case when one does not know about the procedures at all.
We do not do it not because we aren't qualified but because we think an appointed Brahmin can do it more accurately than us. Because, most people may not be even aware of the correct procedures.