I would render John 5:39 this way:
The context makes it unmistakably clear that ἐραυνᾶτε is Jesus' observation about the current occupation of those to whom he is speaking, not his instruction for them to go and be so occupied.
One has to be careful not to coerce the scriptures to support personal points of view. I don't have a copy of Edwyn Clement Hoskyns' work, so I can't read the context of the comment the questioner has given, however, based on the snippet being presented, it looks like the author is using this verse to encourage diligent study of the word.
Surely, the verse is better suited as a cautionary teaching to those who would use the scriptures as a checklist of reasons that eternal life is their due. It seems pretty clear to me that this is the vibe of Jesus' words here -- in regard to the searching of scripture.
Additional Comments - based on the added context for the Hoskyns' quote in the question.
Hoskyns' words capture, precisely, the problem that Jesus saw in the Jewish view of the scriptures.
I call heaven and earth to record this day against you, that I have set before you life and death, blessing and cursing: therefore choose life, that both thou and thy seed may live: That thou mayest love the LORD thy God, and that thou mayest obey his voice, and that thou mayest cleave unto him: for he is thy life, and the length of thy days: that thou mayest dwell in the land which the LORD sware unto thy fathers, to Abraham, to Isaac, and to Jacob, to give them.
-- Deuteronomy 30:19-20
Moses made it clear that the source of life was not the Torah, but GOD: "for HE is thy life".
The purpose of keeping the Law was to advertise the goodness of God.
Behold, I have taught you statutes and judgments, even as the LORD my God commanded me, that ye should do so in the land whither ye go to possess it. Keep therefore and do them; for this is your wisdom and your understanding in the sight of the nations, which shall hear all these statutes, and say, Surely this great nation is a wise and understanding people.
For what nation is there so great, who hath God so nigh unto them, as the LORD our God is in all things that we call upon him for? And what nation is there so great, that hath statutes and judgments so righteous as all this law, which I set before you this day?
-- Deuteronomy 4:5-8
The blessings that came via the Law were supposed to be the empirical evidence that proved God's existence in the midst of his people -- the means by which those outside could see God inside, and thus be drawn to Him.
Eternal life can't be found in the Torah, because it is just a bunch of words in a book, on a scroll or parchment, in this language or that. Eternal life can only be found in THE ONE the words testify about, which Jesus has just informed his listeners, was him.
So, in an indirect, but powerful way (providing additional evidence for the Jewish view of the scriptures), Hoskyns contribution lends support to the indicative alone.