@Benjamin The Red -- In Hebrews 4:14-16, we are told we can approach the throne of grace in boldness with an echo from the OT Leviticus 21:17-21. The throne of God where there is grace and mercy Psalm 89:14. The priestly privilege of access once available to a few is now extended to all Christians. We can now all draw near to God with full confidence and freedom.
Hebrews 4:14-16 has nothing to do with Leviticus 21:17-21. The Law there is explaining which descendants of Aaron (that is, men who are already priests) may not serve at the altar due to defect. The Hebrews passage cites the Ascension of Christ and affirms the fullness of his humanity, stating that we can now receive mercy and grace from God because of Christ. The passage from Leviticus doesn't even mention a throne, unless you keep reading after the passage you cite and read verse 23 which says,
"Only he shall not go in unto the vail, nor come nigh unto the altar, because he hath a blemish; that he profane not my sanctuaries: for I the LORD do sanctify them."
Still yet, it does not say throne, only altar. Perhaps you're trying to equate this word with "mercy seat" and therefore call it the "throne of God"...which is okay. However, that's not what the Hebrews passage is talking about here. In Greek, the word used in Hebrews 4:16 is θρόνος (thronos) which simply means "seat" (not even a particularly nice one, that connotation came later) and not ἱλαστήριον (hilasterion) as St. Paul writes in Hebrews 9:5. Further, in the Septuagint, the word being used here for "altar" is neither θρόνος (which can never mean altar) nor ἱλαστήριον, it's θυσιαστήριον (thusiasterion).
St. Paul in Hebrews 4:14-16 is expressing to us the freedom of grace all Christians receive through the sacrifice of Christ, to whom we draw near (and even partake of at the Eucharist. Even when the priest brings out the Eucharist in the Liturgy he says, "In the fear of God, with faith and love draw near"). This passage has nothing to do with the priesthood of all believers and is not even primarily about the details of Christ's priesthood (for he is setting up to speak about that later in the epistle).
Don't imagine for one moment that i will be put off the point because of your Google-fu.
My goal wasn't to fully exegete Leviticus 21:17-21 or i would have done so. It was merely to highlight the role of the ministerial priests and how people had to go to them and in a sense, through them, to have access to God.
We do not have to go to or through a priest now in any sense. What separates God and man has been dealt with once and for all so that those who call on him, believe in Him can approach Him freely and with boldness.
My Google-fu...you're cute. However, my UBS 4th ed. is insulted.

In one sense what you say is true...in another it is entirely false.
Firstly, it seems that you're still ignoring the parallels between the OT and the NT "priesthood of all believers" and the parallels between the OT Levitical and NT Melchizedek priesthoods. You simply cannot equate the royal priesthood that St. Peter speaks of with the Levitical priesthood of Old. Why? Because he's actually
quoting the Law wherein God is saying this exact same thing to the Israelite people! St. Peter is not claiming that all Christians now function as Christian priests (that is, to offer the Christian sacrifices). All Christians now receive priesthood as did all Israelites of old. The Promises of Israel are now of the Church.
You're absolutely right that Christ has dealt with the problem of sin for humanity, and we can approach God now and receive true forgiveness and reconciliation. That's absolutely 100% Gospel truth. However, you seem to be arguing that this idea is opposed to priesthood. You've created a false dichotomy. Christians still have priests. Priests are how we
become Christians. Show me how many people baptize themselves in the New Testament, or are able to receive the Holy Spirit by their own willing? I'll save you the word searches: None. All of them are baptized by the clergy of the Church, and all must receive the Holy Spirit through the laying on of hands from an Apostle. This is the role of priesthood in Christianity, and it was there from Day One.
So, yes, we all approach God and may attain unity with Him. However, that is done in and through the Church, which provides us with priests who, by the grace of God, baptize and chrismate us, who hear our confessions and pronounce absolution and who officiates at the altar of the Eucharist and are able to say to us faithful communicants: "in the fear of God, with faith and love...draw near."