What Jesus said here was a warning to crowds of people who had pursued him after he had miraculously fed thousands of them with hardly any food. They were interested in another free lunch. This is how he prefaced his warning:
"Ye seek me, not because ye saw the miracles, but because ye did eat
of the loaves, and were filled." John 6:26 A.V.
Then he warned them that it was pursuit of spiritual 'food' that leads to everlasting life that needs to be more important to them than physical food that is put on tables but which only has temporary benefits.
They didn't understand, supposing that when Jesus said, "Labour not for food that perishes..." he meant they had to do some special kind of work, for hard, physical work is needed to produce literal food. They asked him what they had to do, to do the works of God (vs. 28). Jesus' answer?
"This is the work of God, that he believe on him whom he hath sent"
(Ibid. vs. 29)
And how are Christians to believe on Jesus, the One sent by God? There is only one source given to us, and that is the written record of God's word, the Bible. That is what is to be preached by those who follow Christ - the Christ of scripture. To go to any other source of information about Jesus will lead us to 'another Jesus,' a false Jesus who is not the biblical Jesus. Jesus himself warned us (in the New Testament) that many false Christ's would arise and mislead many:
"For many shall come in my name, saying, I am Christ; and shall
deceive many." Matthew 24:5 and see also verse 24, A.V.
With regard to that, you ask what can be said of modern Christianity, which does not show everyone claiming to be a Christian actually following Christ - why would they ignore what Jesus said here? A basic reason why not is that they have no regard for the Bible as being the word of God, which shows us what we are to believe and to do, and how to avoid those false Christs. Here is a quote about that in this church sermon about the Bible:
"What does the Bible say? 'He that believeth and is baptised, shall be
saved; but he that believeth not shall be damned.' [Mark 16:16] If
this be the Word of God, what will become of some of you who have not
read it for the last month? Aye, there are some of you who have not
read it at all. Most people treat the Bible very politely. They have a
small pocket volume, neatly bound, and carry it to their places of
worship. When they get home, they lay it up in a drawer till next
Sunday morning; then it comes out again for a little bit of a treat
and goes to chapel; that is all the poor Bible gets in the way of an
airing. There is dust enough on some of your Bibles to write
'damnation' with your fingers.
I tell you blunt words, but true. What will God say at last? When you
shall come before him he shall say, 'Did you read my Bible?' 'No'.
'I wrote you a letter of mercy; did you read it?' 'No'. 'Rebel! I
have sent you a letter inviting you to me: did you not ever read it?'
'Lord I never broke the seal; I kept it shut up.' 'Wretch!' says God, 'then thou deservest hell. If I sent thee a loving epistle and
thou wouldst not even break the seal, what shall I do unto thee?'
Oh, let it not be so with you. Be Bible readers. Be Bible searchers."
C.H. Spurgeon printed sermon at Exeter Hall, Strand, evening of March 18th, 1855,
Ref. NPSP.15 SP. 198
If that was the state of some Christians attending Church in 1855, what kind of a state do you suppose it is in now? Do we have more Bible readers who are Bible searchers, or less? Therein lies the answer to your second question. For if people will not search the Bible to find the living Christ, they will be 'starving' Christians who will be spiritually dying.