I may have read into this a bit too much but I would like to offer my opinion.
It sounds like tadyatha, literally translated, means "like this, in this way"; relatively translated, the translation can include "spoken like this, the mantra is spoken like this, say the mantra this way, etc"
I would like to offer my own thoughts as to the importance of including tadyatha in recitation.
Firstly I would say that the literal translation is better to use when contemplating the meaning of the mantra during recitation.
If we take Jayarava's three interpretations, which are interpretations of the translation according to the subject of the mantra, to include correct implications, then the meaning of tadyatha is declarative of the subject at hand - the specific mantra or mind protection.
If we consider that reality is like a dream and all manifestations are mind. All sense objects are of mind, etc. We may find that mind protection is influencing the arising of suchness, which is interdependent with the nature of mind, potentiality.
Now it becomes reflexive: as one is reciting the mantra, it is a bit meta to declare that mind is influenced like this, in this way
, nonetheless that is the point of including tadyatha in recitation.
IOW, to say that what you are doing, mind protection, is arising in this way or like this, followed by the specific protection, is implicitly stating the true meaning of reality and the potential of arising to be less or not at all obscured due to its nature. The nature of arising is an illusion, interdependent with the arising of buddha nature, which is a manifestation of the permanence of potentiality. Potentiality is itself not a permanent thing - therefore buddha nature is primordial but not equiprimordial with potentiality.
Since buddha nature includes all aspects of buddhahood as that which is obscured, everything and anything is already those aspects, but is simply appearing according to the net of confusion that makes them appear otherwise. So, tadyatha, in my opinion, means "[it is already] like this..."
for instance, "Homage to the Bhagavan Bhaisajyaraguru Vaiduryaprabharaja, a tathagata, an arhat, a samyaksambuddha... [the universe is already existing in a perfect way, according to the power of the potentiality of the buddha, however, to a confused being like me, it appears not to be so, thus I praise you and pray that] in this way (by reciting the following) [reality may manifest according to the following request]: throughout the entire universe, do away with the pain of illness, do away with the pain of spiritual illness, do away with the pain of illness, all throughout. I offer this prayer to you [making it inseparable from reality]."
That's my opinion and the way I've come to understand tadyatha, but who knows what the Buddha meant
Much love to you all