A lay person can expel through adjuration. There are many factors involved that can’t be ignored. The disposition of the person possessed, the circumstances that allowed the possession, the possessed persons state once the faculties are released. An exorcism not followed up with proper care can place the soul in a worse condition than before.
Anyone can command an evil spirit to leave in the Name of Christ but what if the demon is there under the command of another who has rights over it? Perhaps the adjuration of the lay exorcist are eneogh to compel the demon but it can’t act because it would violate an oath to another. These aren’t meaningless matters for good or evil spirits. Only the Church has rights over boundaries set up by evil spirits, or doors permitting domain, or evil oaths between persons. A priest wielding the authority Jesus handed over to the Church to command demons removes any appeal to rightfull dominion an evil spirit may claim. A demon may claim that it can’t leave because of an oath. It may even be true. The lay exorcist can only appeal to the authority Christ handed to the Church. The authority of the Apostles invested in the Bishop wielded by the Priest has power over any evil claims.