This will not be a complete answer, but ...
In Greek, the whole prayer begins:
’’Ετι προσφέρομέν σοι την λογικην ταύτην καì αναίμακτον λατρείαν ...
Again we offer unto you this rational and bloodless worship ...
The key words are λογικην (logikin) and λατρείαν (latreian).
λογικην can mean "logical", but I have also seen it translated "reasonable", "rational", "spiritual" and "noetic". I might be drawing the bow too far, here, but I suppose it could also be translated "according to (the) Logos" or Christ himself -- no doubt a better Greek-speaker will correct me.
λατρείαν means adoration, worship.
Remember that in the ancient world all worship was made by sacrifice, usually bloody sacrifice. So many of the words tied to worship in ancient languages are related to the concept of offering up or burning whole or letting blood. This is also highlighted by the appearance of λογικην so close to αναίμακτον -- "unbloody", "without shedding blood".
By contrast, the Divine Liturgy is offered in spirit and truth, not in blood. The worship we give to the Creator is offered by our intelligent and noetic self, not only according to flesh as a pagan sacrifice.
So, in at least that way, the worship and adoration that is offered to God in the liturgy is λογικην.
"Spiritual" and "noetic" are looser translations of λογικην than "reasonable" and "rational", but I think they fit better. I don't think St John's words are meant to be advertising how obvious and logical the sacrifice of the Divine Liturgy is -- as if it could be arrived at by the application of reason alone. Nevertheless, it is reasonable.
I don't doubt that, like everything else in the Divine Liturgy, there is a fuller and deeper explanation than the surface-level one I have offered.
Hope this helps.