Did only bishops preach in the early Church?

Bishops were not the only ones who preached in the early church.

As an example take the longer section of scripture stretching from Acts 6-7. In those words the context is clear: The ministry of the word was being neglected because the apostles had to "wait on tables." (“διακονεῖν τραπέζαις” (Acts 6:2 UBS5-T))

In response to this need, the apostles drew up a list of names to be deacons. And, in this context, we see what deacons were. They were called on to take care of the administrative, service work so that time could be freed up for the apostles to pray and study and share God's word.

From this we would think that these deacons had no part in the ministry of the word. Then Stephen gets up and preaches a long and amazing sermon proving that Jesus is the Christ. And, far from condemning his actions, all the apostles and the rest of the Christian Church are both approving of what he did and were amazed by it.

We see that Stephen validly preached a sermon both because of the lack of pushback against his sermon. But we also see it within the greater context of the book of Acts. Whenever there is an important sermon in the book of Acts, the speaker signals the sermon with the words, "Men, brothers." (“Ἄνδρες ἀδελφοί” (Acts 13:15 UBS5-T)) This pattern shows up 18 times within the book of Acts. Peter uses those words. Paul uses those words. And Stephen uses those same introductory words.

Much more could be said here. But there is more than enough evidence within the early church included in the book of Acts to see that the statement, "In the beginning it was the Bishops alone who preached" stands in direct opposition to the date and facts we have in the book of Acts.

Did only bishops preach in the Early Church?

The short answer seems to be no.

The deacon Philippe of Acts 6 was equally an evangelist in the sense that he preached the Gospel of Christ. The Catholic Encyclopedia explains the meaning of the term evangelist as follows:

In the New Testament this word, in its substantive form, occurs only three times: Acts 21:8; Ephesians 4:11; 2 Timothy 4:5. It seems to indicate not so much an order in the early ecclesiastical hierarchy as a function. The Apostles, indeed, were evangelists, inasmuch as they preached the Gospel (Acts 8:25; 14:20; 1 Corinthians 1:17); Philip likewise was both a deacon (Acts 6:5) and an evangelist (Acts 8:4-5; 8:40; 21:8); in like manner was St. Timothy exhorted by St. Paul to do the work of an evangelist (2 Timothy 4:5).

From the various statements contained in the New Testament, we may gather with some probability that evangelists were travelling missionaries, occasionally solemnly set apart, as seems to have been the case with Sts. Paul and Barnabas (Acts 13:1-3), to go about and preach the Gospel, yet sometimes with a settled place of abode, as Philip at Cæsarea, and Timothy at Ephesus. They were endowed with a special charisma to preach to those unacquainted with the Christian Faith and pave the way for the more thorough and systematic work of the pastors and teachers. But their office, as such, seems to have extended no further, so, for instance, we understand from Acts 8:4 sqq., that Philip, who preached successfully in Samaria and baptized many, was not qualified to impart the Holy Ghost to the converts (verse 14). Accordingly, St. Paul, in his list of the gifts bestowed by Christ for the edification of the Church, Ephesians 4:11 (in 1 Corinthians 12:28, they are omitted), mentions the evangelists in the third place, only after the Apostles and the Prophets. In the writings of the Apostolic Fathers, no reference is made to evangelists; travelling missionaries are sometimes called "apostles", sometimes also, as in the Didache, they are styled "teachers".

Deacons obviously shared in the Sacred Work of the Apostles.

St. Clement of Rome (about A.D. 95) clearly describes the institution of deacons along with that of bishops as being the work of the Apostles themselves (Ep. Clem., xlii). Further, it should be noted that ancient tradition limited the number of deacons at Rome to seven (Eusebius, Church History VI.43), and that a canon of the council of Neo-Caesarea (325) prescribed the same restriction for all cities, however large, appealing directly to the Acts of the Apostles as a precedent. We seem, therefore, thoroughly justified in identifying the functions of the Seven with those of the deacons of whom we hear so much in the Apostolic Fathers and the early councils. Established primarily to relieve the bishops and presbyters of their more secular and invidious duties, notably in distributing the alms of the faithful, we need not do more than recall the large place occupied by the agapae, or love feasts, in the early worship of the Church, to understand how readily the duty of serving at tables may have passed into the privilege of serving at the altar. They became the natural intermediaries between the celebrant and the people. Inside the Church they made public announcements, marshaled the congregation, preserved order, and the like. Outside of it they were the bishop's deputies in secular matters, and especially in the relief of the poor. Their subordination and general duties of service seem to have been indicated by their standing during the public assemblies of the Church, while the bishops and priests were seated. It should be noticed that along with these functions probably went a large share in the instruction of catechumens and preparation of the altar services. Even in the Acts of the Apostles (8:38) the Sacrament of Baptism is administered by the deacon Phillip.

If you are asking if only the bishops preached at Mass, then I would say there is a dearth of information on that subject matter, but it is not impossible.