For the earliest church, we have no definite information. So Lipscomb's statement must be rejected as tendentious (the author justifies his own church's tradition of vocal music only) when he claims:
Neither [Paul] nor any other apostle, nor the Lord Jesus, nor any of
the disciples for five hundred years, used instruments.
In fact, Jesus himself spoke of his own metaphorical flute playing in Mt. 11:17:
17 ‘We played the flute for you, but you did not dance, we sang a
dirge but you did not mourn.’
This, of course, it not convincing evidence, but it does cause one to wonder why Jesus would refer to himself playing a metaphorical flute if he intended to forbid his disciples from playing real ones. A more definite indication is the fact instrumental music was an important part of Jewish worship in the first century. During this time the Christian church was largely Jewish and even many Jewish priests joined the church.
The word of God continued to spread, and the number of the disciples
in Jerusalem increased greatly; even a large group of priests were
becoming obedient to the faith.
In addition, Acts 21 indicates that the Temple itself was still a key center of religious worship among Jewish Christians, who were the majority in Jewish lands as well as a substantial demographic group in the church throughout the Roman Empire. While the Temple of Jerusalem still stood, instrumental music was an [integral part of Jewish worship] as indicted in the psalms that speak of making a "joyful noise" unto the Lord with stringed instruments, horns, etc. This was true for synagogues as well as the Temple, and there is no indication that Christian congregations at this point differed dramatically from their Jewish counterparts in worship style.
However, the Jewish attitude toward instrumental music changed dramatically after the Temple fell, as it was felt that mourning rather than joy was the appropriate religious attitude. Meanwhile, once the Jerusalem church was dispersed, Christian liturgical tradition increasingly diverged from Jewish customs. The attitude soon evolved that instrumental music was "of the body" or of "the old covenant" and that only vocal music was truly spiritual. This attitude probably did not take form suddenly, nor was it universal, but it was well established by the fifth century.
Paul James-Griffiths writes in an article on the History of Music in the Church:
The majority of Church Fathers between AD 100 and 500 did not accept
the use of musical instruments in church and the Christians worshipped
God with psalms, hymns and spiritual songs in a chanting fashion...
Apart from the rejection of musical accompaniment during worship
because they regarded it as being from the Old Covenant, they were
also defensive about the possible influences of pagan music creeping
into the Church and leading it astray.
However, some Church Fathers, while preferring vocal music, clearly allowed instruments. For example Clement of Alexandria - Instructor 2:4
And even if you wish to sing and play to the harp or lyre, there is no
blame. You shall imitate the righteous Hebrew king in his thanksgiving
Conclusion: The statement quoted in the OP is inaccurate. The earliest church probably did use musical instruments, especially in Jewish-Christian congregations, until sometime after 70 CE. After that, the tradition to avoid musical instruments in the period discussed was not universal, although it was fairly well established.