They must intend to do what the Church does. If there is no intention as in a play then it is not baptism.

You must intend to DO what the Church DOES.  That's why atheists can validly baptize.  Apart from that, the details of the story are likely embellished somewhat and the account cannot be considered completely reliable.  So whatever you want to make it of it, it's not proof of anything, poche.  Try actually making a theological argument.

So St. Genesius had an outpouring of grace, converted from his paganism, and asked to be baptized, trusting in God to provide what was needed.  Is it your position that God gave St. Genesius the grace of conversion but neglected to give the grace necessary to the other actor to baptize with the basic intention of doing what the Church does?   Bear in mind that this is the same God who said "Unless a man be born again of water and the Holy Ghost, he shall not enter the Kingdom of God", and again "He that believeth and is baptized, shall be saved: but he that believeth not shall be condemned", and elsewhere: "Ask, and it shall be given you: seek, and you shall find: knock, and it shall be opened to you. For every one that asketh, receiveth: and he that seeketh, findeth: and to him that knocketh, it shall be opened. Or what man is there among you, of whom if his son shall ask bread, will he reach him a stone? Or if he shall ask him a fish, will he reach him a serpent?  If you then being evil, know how to give good gifts to your children: how much more will your Father who is in heaven, give good things to them that ask him?"

One of the biggest problems with those who deny EENS is a lack of faith in Divine Providence.  Not a single hair falls from your head without God willing it.  Not a single drop of water falls from the baptizing vessel without God directing it.

This baptism was simulated and Genesius' conversion came after the baptism not as a result of it

This baptism was simulated and Genesius' conversion came after the baptism not as a result of it

Not according to the narrative I read.  What I read indicated that he received a miraculous grace of conversion just before the Baptism and truly willed and desired to receive it by the time it was administered.  Again, it goes to show you that dubious narratives such as the account of his martyrdom cannot be used to do theology.

Water baptism is the only true baptism and it's necessary for salvation, which is why God even grants (baptized) atheists the ability to validly water baptize a willing person in times of imminent danger to that person's life.

In case of necessity, God permits even unbaptized atheists to baptize.

From Trent's Catechism:


Ministers In Case Of Necessity

Those who may administer Baptism in case of necessity, but without its solemn ceremonies, hold the last place;
and in this class are included all, even the laity, men and women, to whatever sect they may belong. This office
extends in case of necessity, even to Jєωs, infidels and heretics, provided, however, they intend to do what the
Catholic Church does in that act of her ministry. These things were established by many decrees of the ancient
Fathers and Councils; and the holy Council of Trent denounces anathema against those who dare to say, that
Baptism, even when administered by heretics, in the name of the Father, and of the Son, and of the Holy Ghost,
with the intention of doing what the Church does, is not true Baptism.

And here indeed let us admire the supreme goodness and wisdom of our Lord. Seeing the necessity of this
Sacrament for all, He not only instituted water, than which nothing can be more common, as its matter, but also
placed its administration within the power of all. In its administration, however, as we have already observed,
all are not allowed to use the solemn ceremonies; not that rites and ceremonies are of higher dignity, but
because they are less necessary than the Sacrament.