There are three questions about John 7:4, 10 here that should be separated but I will take them all here.
1. Secret or Private?
The operative word here is κρυπτός (kruptos) which means (BDAG) "secret" or, "unknown", or "hidden". Thayer adds, ""concealed", "private", "privately". Certainly, in Matt 6:4, 6, 18, "private" would be a better translation. I would also suggest that "private" is the best in John 7:4, 10 for the following reasons:
- When people went to the three annual festivals in Jerusalem, they almost always travelled in groups or caravans both for companionship and security.
- Indeed, a number Ps 120-134 are each entitled, "Song of Ascents" because these were designed to be sung when going "up" (1Ki 12:27, 28) to Jerusalem.
- When travelling from the region of Galilee to Jerusalem, Jews invariably avoided travelling through Samaria and thus travelled the longer route west of the Jordan. If Jesus travelled through Samaria (Compare John 4:4), alone, he would arrive in Jerusalem unnoticed.
Ellicott reaches the same conclusion:
Not openly, but as it were in secret—i.e., not with the usual company.
Judging from His practice at another time (John 4:4), He would go
through Samaria, while the caravan would go on the Eastern side of the
Jordan.
2. Why private?
Jesus' activities had aroused much discussion and a rising tide of official hostility. If Jesus had arrived in Jerusalem openly, he may have been arrested before arriving and thus prevented from accomplishing his work before it was finished. Thus He said to His brothers, "My time has not het come" (John 7:6). Note the comments of Benson:
With as much privacy as possible; and that probably rather for fear of
giving offence than of receiving injury: he would not unnecessarily
provoke the government, which his being accompanied with a multitude
of people would have done. And this suggests another reason for his
delay. Had he taken this journey at the usual time, the multitudes who
were on the road would have gathered round him, and accompanied him to
Jerusalem, and at once have excited the notice, and provoked the
malice and envy of his enemies; he therefore did not set out till the
greater part of the people were gone, and then went up as privately as
possible, neither preaching nor working miracles by the way.
3. Did Jesus Lie to the Brothers?
John 7:8 as most version have it ("I am not going up … ") appears to make Jesus either change His mind or lie about his intentions about going to the feast. Let us examine the Greek more closely:
NA27/NA28/UBS4/UBS5 etc all have: ἐγὼ οὐκ ἀναβαίνω (= I am not going up) but all have this as quite uncertain and rate this as {C}. They also list an almost equally possible reading: ἐγὼ οὔπω ἀναβαίνω (= I am not yet going up). This is the reading preferred by NA4, W&H, Byzantine text, Orthodox text, Majority text, TR, etc, and is attested quite early, eg, P66 & P75. For obvious reasons, I prefer this latter reading. The “yet” supports what actually happened - Jesus did go but a day or so later.