Here are some important verses in John 6:
1-6 After this Jesus crossed the Lake of Galilee (or Lake Tiberias), and a great crowd followed him because they had seen signs which he gave in his dealings with the sick. But Jesus went up the hillside and sat down there with his disciples. The Passover, the Jewish festival, was near. So Jesus, raising his eyes and seeing a great crowd on the way towards him, said to Philip, “Where can we buy food for these people to eat?” (He said this to test Philip, for he himself knew what he was going to do.)
7 “Ten pounds’ worth of bread would not be enough for them,” Philip replied, “even if they had only a little each.”
8-9 Then Andrew, Simon Peter’s brother, another disciple, put in, “There is a boy here who has five small barley loaves and a couple of fish, but what’s the good of that for such a crowd?”
10a Then Jesus said, “Get the people to sit down.”
10b-12 There was plenty of grass there, and the men, some five thousand of them, sat down. Then Jesus took the loaves, gave thanks for them and distributed them to the people sitting on the grass, and he distributed the fish in the same way, giving them as much as they wanted. When they had eaten enough, Jesus said to his disciples, “Collect the pieces that are left over so that nothing is wasted.”
13-14 So they did as he suggested and filled twelve baskets with the broken pieces of the five barley loaves, which were left over after the people had eaten! When the men saw this sign of Jesus’ power, they kept saying, “This certainly is the Prophet who was to come into the world!”
15 Then Jesus, realising that they were going to carry him off and make him their king, retired once more to the hill-side quite alone...
... Some other small boats from Tiberias had landed quite near the place where they had eaten the food and the Lord had given thanks. When the crowd realised that neither Jesus nor the disciples were there any longer, they themselves got into the boats and went off to Capernaum to look for Jesus. When they had found him on the other side of the lake, they said to him, “Master, when did you come here?”
26-27 “Believe me,” replied Jesus, “you are looking for me now not because you saw my signs but because you ate that food and had all you wanted. You should not work for the food which does not last but for the food which lasts on into eternal life. This is the food the Son of Man will give you, and he is the one who bears the stamp of God the Father.”
Context is a wonderful thing. When Jesus talked about bread, he was referring BACK to the miracle of the previous day, and the one that had brought so many followers to him. None of what he said would make sense to ANY listener as referring to the Last Supper, which was a few years away...
Yes, context is a wonderful thing. What was left out of your context was Jesus explicitly speaking of the manna which the ancestors of these men ate in the desert. The manna was miraculous, a gift from God for their physical sustenance.
That discourse is much more salient to the discussion of the reality of Christ’s presence in the Eucharist than the miracle of feeding the five thousand which, like the manna, was a mere precursor to the real miracle.
There is so much in that passage that supports the Eucharist and nothing that contradicts the Catholic belief or disproves it either.