To answer this question of whether Jesus was truly acting as the savior requires an understanding of Tabernacles as well as burnt offerings. Once that is obtained, the answer to the question is given by Christ himself in the passages cited by the OP.
The Feast of the Tabernacles is the wedding feast, it is a prophecy of the marriage supper of the Lamb. It is a parable for all creation, for in the first seven days there is feasting and celebration, but on the eighth day, the marriage is consummated. During the week, we go through the seven days and then cycle back to the first day. But to reach the eighth day is to escape this cycle. So the eighth day is always symbolic of eternity, of escaping the seven day loop. Only those entering eternity in the marriage supper of the Lamb are allowed into the eighth day - the great day - while everyone else is sent away from the wedding feast.
Thus Tabernacles marks the union of God (in the form of Christ) and his church (consisting of both jew and greek), in eternity.
In the Law of Moses, Israel acts out this divine union by performing various symbolic rituals. Without going into details of all the performance, here are some key points:
Israel is a type for the church, all those who see God. Thus all Israel participates as the bride.
Jesus is the groom. The groom is represented by the burnt offering and by the high priest. But these are only types for the groom.
living in booths for seven days is a type for our life in the flesh.
The giving of burnt offerings is a type for dedicating ourselves wholly to Christ and simultaneously it is a type for his complete dedication to the Father. This mutual dedication is itself a type for the union of God and man. When we seek him, we offer him as the sacrifice, but when we are like him, we offer ourselves as living sacrifices.
During the seven days of the feasts, the garments of the priests are drenched in oil and burned to provide light for the people. This is a type of Christ expending his flesh in order to be the light of the world but also of him giving teachings to the bride.
But in the eighth day the booths are burned -- our own flesh is destroyed on the eighth day, which is when we enter into eternity, into union with him.
In the eighth day, there is a procession in which the paradise apples (in the form of pomegranates) are smashed, as the congregation symbolicly promises to never judge for themselves what is good and evil. This is a type for the transformation of our minds as we are made one with Christ.
Was Jesus performing his duty as the groom?
Now Jesus was fulfilling his role as the groom, because
- he was the burnt offering, fully dedicated to God. We know that his offering was accepted because he was transfigured by the Father (most likely on a different Tabernacles, as Peter wanted to build booths then).
And a voice came from the cloud, saying, “This is my Son, my Chosen
One. Listen to him!” [LEB]
- He was giving his teachings to the bride
- He had come to tabernacle among man.
Was Israel performing her duty as the bride?
The bride was playing the whore. The jews were refusing to follow him and to feast with him, but were following their own ideas and idols. Thus their burnt offerings were not accepted!
How do we know that their burnt offerings were not accepted? We have in Hosea 6.6:
Because I desire steadfast love and not sacrifice, and knowledge of
God rather than burnt offerings. [LEB]
It was more important, in God's eyes, for the jews to know Jesus than to perform the burnt offerings. Of course the best was to know Jesus and perform the burnt offerings in Spirit and Truth. But as that was not on the table, he sent his Son to call the bride to repentance even on Tabernacles. He was giving light (teaching) during the feast.
How Jesus responds to save the bride
Being sent by the Father, Jesus began to call out, as per the parable of the wedding feast [which was spoken with Tabernacles in mind and most likely on Tabernacles]:
And continuing, Jesus spoke to them again in parables, saying, “The
kingdom of heaven may be compared to a man—a king—who gave a wedding
celebration for his son. And he sent his slaves to summon those who
had been invited to the wedding celebration, and they did not want to
come. Again he sent other slaves, saying, ‘Tell those who have been
invited, “Behold, I have prepared my dinner; my oxen and fattened
cattle have been slaughtered, and everything is ready. Come to the
wedding celebration!” ’ But they paid no attention and went away—this
one to his own field, that one to his business. And the others,
seizing his slaves, mistreated them and killed them. And the king was
angry and sent his troops and destroyed those murderers and burned
their city. Then he said to his slaves, ‘The wedding celebration is
ready, but those who had been invited were not worthy. Therefore, go
out to the places where the roads exit the city and invite to the
wedding celebration as many people as you find.’ And those slaves went
out into the roads and gathered everyone whom they found, both evil
and good, and the wedding celebration was filled with dinner guests.
But when the king came in to see the dinner guests, he saw a man there
not dressed in wedding clothes. And he said to him, ‘Friend, how did
you come in here, not having wedding clothes?’ But he could say
nothing. Then the king said to the servants, ‘Tie him up hand and foot
and throw him into the outer darkness. In that place there will be
weeping and gnashing of teeth!’ For many are called but few are
In other words, Jesus could have said "screw Israel, they are not feasting with me, their groom, during the wedding feast, so let them offer their vain sacrifices and be locked out of the eighth day. I will feast with the few who know me already."
But from his mercy as messiah, he decided to intervene by calling his bride to him.
When is the right time to save the bride?
Now the original poster, like the pharisees, is complaining that this is the wrong day to save Israel! But Jesus answers both the pharisees accusing him and the OP with this statement:
If a man receives circumcision on the Sabbath so that the law of Moses
would not be broken, are you angry with me because I made a whole man
well on the Sabbath? Do not judge according to outward appearance, but
judge according to righteous judgment!” [LEB]
In other words, don't worry that I am saving souls on the wrong day, bringing them into the true wedding feast and the true rest! Do not judge according to the outward form, but judge according to what is right!