This understanding of the dialogue was not uncommon among the Fathers of Church. For example St. John Chrysostom wrote:
If the Holy Spirit were not in our Bishop [referring to Bishop Flavian of Antioch] when he gave the peace to all shortly before ascending to his holy sanctuary, you would not have replied to him all together, And with your spirit. This is why you reply with this expression…. reminding yourselves by this reply that he who is here does nothing of his own power, nor are the offered gifts the work of human nature, but is it the grace of the Spirit present and hovering over all things which prepared that mystic sacrifice. (Pentecost Homily)
The first obvious change will be the response to “The Lord be with you”, which will be “And with your spirit”. This brings us into line with all the other major languages where “et cum spiritu tuo” is translated literally, for example “con il tuo spirito” in Italian. This is an ancient mystical response to a blessing-greeting, referring either to the priest’s angelic guardian or the presence of the Holy Spirit in the priest or, as is more likely, the spirit within each us.
This is why this blessing-greeting with its response is only given by the ordained and not by lay people, for example when a lay person leads a “Communion service”.