Does the mention of Sheep Gate in John 5:2 give a lead to explanation for the allegory of Camel and Eye of the Needle?
Summary, John 5:2 and Matthew 19:24 are unrelated. John 5:2 does not lead to explanation for the allegory of the Camel and Eye of the Needle.
Let's look at the passages.
1 Some time later, Jesus went up to Jerusalem for one of the Jewish festivals. 2 Now there is in Jerusalem near the Sheep Gate a pool, which in Aramaic is called Bethesda and which is surrounded by five covered colonnades. 3 Here a great number of disabled people used to lie—the blind, the lame, the paralyzed and they waited for the moving of the waters. 4 From time to time an angel of the Lord would come down and stir up the waters. The first one into the pool after each such disturbance would be cured of whatever disease they had. 5 One who was there had been an invalid for thirty-eight years. 6 When Jesus saw him lying there and learned that he had been in this condition for a long time, he asked him, “Do you want to get well?”
In this passage "the Sheep Gate" is a location providing the setting for this miracle.
21 Jesus answered, “If you want to be perfect, go, sell your possessions and give to the poor, and you will have treasure in heaven. Then come, follow me.”
22 When the young man heard this, he went away sad, because he had great wealth.
23 Then Jesus said to his disciples, “Truly I tell you, it is hard for someone who is rich to enter the kingdom of heaven. 24 Again I tell you, it is easier for a camel to go through the eye of a needle than for someone who is rich to enter the kingdom of God.”
25 When the disciples heard this, they were greatly astonished and asked, “Who then can be saved?”
26 Jesus looked at them and said, “With man this is impossible, but with God all things are possible.”
Here Jesus is referring to an actual camel and an actual needle using humor to make the point that wealth is a barrier to entering the kingdom of God. This was a concept that was counter to the culture in Jesus' time as demonstrated by the disciples response "they were greatly astonished". They believed that wealth was an indicator of God's favor.
Poor Bible teaching over the years has tried to read other explanations into Scripture by inventing a "Needles Eye Gate". There is no archeological, textual, or contextual evidence for such a gate. The erroneous teaching says:
The "Eye of the Needle" has been claimed to be a gate in Jerusalem, which opened after the main gate was closed at night. A camel could not pass through the smaller gate unless it was stooped and had its baggage removed. The story has been put forth since at least the 15th century and possibly as far back as the 9th century.
This invention allows the camel to get through the gate if it is "humbled" - has no baggage and crawls. This completely misses the point of the passage.
The original question posits a theory which includes a "Camel Gate".
What if the so-called Camel Gate was designed NOT to allow camels to pass, modeled after the cattle grid ? That would prevent smuggling of things to and out of the city in the night . It is also possible that the Sheep Gate in John 5:2 had similar purpose i.e. NOT allowing the sheep to pass, so that genuine travelers got the right of way .
There is no Camel Gate in John or Matthew.
Nehemiah 3:1 - 32 details the building of the gates - there is no Camel Gate listed.