And this is the will of him that sent me that every one which seeth the Son
and believeth on Him...
It seems the qualification is that God's will in this case refers to those who "seeth" the Son (as well as believing in Him), should have eternal life. This verse says nothing about ONLY those who see and believe in Him having eternal life, but that all who DO see and DO BELIEVE may have eternal life. The verse does not GUARANTEE eternal life even to those who see and believe. God wills that every soul may have eternal life. If God WILLED for every man, woman, and child to believe in His Son, then every man, woman, and child from the beginning of time would be aware of the Trinity, which is simply not true, UNLESS every man, woman, and child, regardless of origin, upbringing, religion, etc., will receive grace at the moment of death to "see" and "believe" in Jesus Christ. Thus, the harlots and publicans who are entering heaven before you and I are those who, despite knowing not Christ, receive His grace at death. We (you and I) already have that grace. Thus, "to whom much is given, much is expected".
Additionally, this grace offered to those who know nothing of Christ can be rejected just as you and I can reject His grace at our own last breath.