The difficulty in answering this question is that the liturgy of the early Church was largely oral in nature, and so we contain virtually no records from the early Church itself regarding what it exactly did in its own liturgy. Contrary to the author who posted the quote you use, if we have evidence of the Lord's Prayer in use even several centuries later this is a very good sign to the tradition's ancient history (and an even better sign if the Lord's Prayer is spoken about with great reverence ). It is from the late third century (more so the early fourth century) and onward that we gather most of our knowledge about the Church in the first two centuries or so. I would also add that when first and second century Church Fathers were speaking about theological matters, it is widely assumed that the common division we now have between 'liturgy' and 'personal practice' was quite less potent. For these people church wasn't a specific place they went to to partake in a fully organized procedure; it was a way of life itself. When people met, it was an embodiment of individual holy life in communion (particularly through the partaking of the Eucharist). I think it is safe to say that for these people, their faith and theological understandings were meant to be carried out in the personal life, and thus logically and presumably in the preceding liturgical life. That being said, we can now analyze the evidence for the age of the Lord's Prayer.
Didache
Perhaps the earliest mention of the Lord's prayer outside of the Gospels is found in the Didache. The didache is an early Church order, thought to be perhaps the first 'catechism' of the Church. It states the following:
But let not your fasts be with the hypocrites (Matthew 6:16) for they fast on the second and fifth day of the week; but fast on the fourth day and the Preparation (Friday). Neither pray as the hypocrites; but as the Lord commanded in His Gospel, thus pray: Our Father who art in heaven, hallowed be Your name. Your kingdom come. Your will be done, as in heaven, so on earth. Give us today our daily (needful) bread, and forgive us our debt as we also forgive our debtors. And bring us not into temptation, but deliver us from the evil one (or, evil); for Yours is the power and the glory for ever. Thrice in the day thus pray.
The Didache is thought to have been written anywhere between the mid to late first century. So we have not only references to the Lord's Prayer in Church Father teachings (which shall be revealed later) but in what Eusebius considers to be a canonical catechism of the early Church. In any case, it illustrates that the Lord's Prayer is an instituted tradition already existing within the Church. This is said because it is rather clear that the writer(s) of the Didache were referring to the idea that Christ commanded that the Church pray the Lord's prayer. This is thus not some tradition that is being instituted in light of a new principle, but rather confirmed in light of what was believed to be a preexisting commandment from the Lord himself.
Tertullian
Tertullian's apostolic work was around the time the Didache came into being, and so it is likely that his thoughts on certain matters were influenced by this writing. Tertullian shares thoughts on the Lord's Prayer, which only further confirms the Didache's stance and suggests the Prayer's grounded place in the Christian culture at the time.
Since, however, the Lord, the Foreseer of human necessities, said separately, after delivering His Rule of Prayer, Ask, and you shall receive; and since there are petitions which are made according to the circumstances of each individual; our additional wants have the right— after beginning with the legitimate and customary prayers as a foundation, as it were— of rearing an outer superstructure of petitions, yet with remembrance of the Master's precepts.
Although here Tertullian is arguing that personal prayers can be 'superadded' to what he calls 'customary prayers,' it is the fact of the 'customary prayers' that is of interest. Tertullian acknowledges some existing custom of the Lord's Prayer (as stated in the title of the segment itself) within the liturgy or nature of how Christians at the time prayed (again, keep in mind there was little difference between the two during this time). Besides this, in the previous segments, Tertullian spends an ample amount of space going through several specific lines of the Lord's Prayer. So we see Tertullian both acknowledging the Lord's Prayer as an existing custom in the Church, and expounding on its meaning in his own terms. This suggests that the Prayer was a very meaningful tradition, and as a 'custom' likely something that was precisely liturgical.
Cyprian
Cyprian was born around 200 A.D and became the Bishop of Carthage. Cyprian, following Tertullian, supported the Lord's Prayer as well in his writings. This is particularly important for it reveals how the Church, in regards to the catechism of the didache, did not face divide in itself over a linear sequence over the doctrine of the Lord's Prayer, as so many doctrines cause.
Let us therefore, brethren beloved, pray as God our Teacher has taught us. It is a loving and friendly prayer to beseech God with His own word, to come up to His ears in the prayer of Christ.
Here again we find the same support of what appears to be a tradition within the Church.
The Liturgy Within
I would hold that on top of the evidence we have supporting an early presence and practice of the Lord's Prayer in the body of the Church, the texts themselves suggest specifically liturgical practice of the Lord's Prayer.
Firstly, in considering the Didache, we must recognize the purpose for which it was written. The Didache was written, as stated above, as a catechism. It should be made known that the orders regarding the Lord's Prayer are directly in between and associated with baptism and the Eucharist. These two practices are undeniably liturgical in nature, so the fact that the Lord's Prayer and an explanation of it is so associated with the two Sacraments suggests that it is in fact a liturgical tradition. Beyond this, it should be stated that the purpose of the Didache is not a spiritual guide for the individual but rather a general order for the community (aka the communion of Christians, or the liturgy).
Secondly, in considering Tertullian's writings on the Lord's Prayer, we find that he considers it to be a 'customary' prayer, or a foundation on which he states all 'personal' prayers rest. This suggests that the Lord's Prayer is precisely not entirely personal by nature, but rather grounded in the general order of Christ that his Church pray like he, and through his words. The contrast Tertullian purposely makes between the Lord's Prayer and 'subadded' prayer from the individual is indicative that the Lord's Prayer is more central not only as a 'personal' prayer but as a prayer that is shared and held by all members of the Church, and thus undoubtedly expressed whenever there is communion between the said members.
Final Thoughts
If we take into account how early the Didache came into being (between the mid to late first century), how firmly this catechism was affirmed by following bishops and Church Fathers, I think we are left with a good reason to believe the Lord's Prayer was instituted in the very early Church, likely by the disciples themselves. Recall that the accounts of the Gospels tells us that Christ stated 'pray like this' to his disciples before starting the Lord's Prayer. This commandment language is similar to His institution of other sacraments, such as baptism and the Eucharist. I think it is unlikely that there existed a gap in which the Church didn't think the Lord's Prayer should be recited, not only personally but publicly. While we don't have many examples of liturgical practice including the Lord's Prayer during the very early Church, such is only because we don't have many written examples of liturgical practices period. Looking at surrounding texts however brings more information for the historian, and what this information tells us is that the Lord's Prayer is almost as sure in presence of the early Church as the Lord is himself. Based on the facts that we have elsewhere about the nature of liturgy, I think it is safe to say that these texts were calling for the Lord's Prayer not only in 'personal' communion but in 'public' communion. The main reason I say this is based on the details that suggest this in the texts themselves and the valid reason we have to not assume otherwise (this was stated in the introduction but not formally expounded upon). There is little reason to believe the liturgies at the time would not include (for the majority of the time at least) something so obviously paramount as the Lord's Prayer.
