The question and the current answers seem to be using terms like "Jews", "Israel", and "Chosen" very ambiguously, and their meanings are confused and conflated.
Before one can discuss "Chosen People", one really needs to understand the background details.
God promised various conditional and unconditional blessings to Abraham.
These blessings were inherited by his son Isaac, and then by his son Jacob.
God gave the name Israel to Jacob, and his descendants are known as Israelites.
Jacob's twelve sons each became one of the twelve tribes of Israel.
His son Judah inherited the royal line (including the "sceptre" promise of the ultimate king, the Messiah).
His son Joseph inherited the physical blessings and name.
Later, his son Levi's tribe was given the priesthood.
Following King Solomon's reign, the tribes separated into two nations.
The northern part retained the name "Israel" and the birthright blessings because it included Joseph.
The southern part was called "Judah" because it was mostly the tribe of Judah, along with the remnant of Benjamin and some of Levi, and it retained the royal line.
The people of Judah became known as Jews.
That name is first used in 2 Kings 16:5, where Syria and Israel are fighting a war against the Jews, clearly demonstrating the separation of Judah from Israel.
In the 20th century, a nation known as "Israel" was created, its citizens being known as Israelis (not Israelites).
These people are mostly from the tribe of Judah (Jews), and are not the full twelve tribes of Israel.
History is written by the winners, and so most ancient documents tend to reflect only the great deeds of the people that wrote them.
As a historic document, the Bible is unique in that it records everything, warts and all, from the glory of David and Solomon's empire through the atrocities of various decadent kings, and the eventual captivity of Judah and Israel.
The Bible records that whenever Israel or Judah had a good king, one that respected God's laws, the nation followed his example and the nation prospered.
And whenever it had a king that disrespected God's laws, practicing idolatry, the people followed his example and the nation suffered.
The Pharisees, as mentioned in the Greek scriptures 2000 years ago, were well aware of this historical relationship and made it their mission to ensure that the people of Judah follow God's laws, thereby making the nation worthy of God's blessing.
They expanded God's laws to make it much more obvious whether one was following them or not (it was these expanded laws that Jesus was frequently accused of breaking).
A modern example of one of these rabbinical fences is how traffic intersections are regulated.
Yield-signs are installed to indicate that the crossing road has right-of-way so that cars don't collide (God's law).
Stop-signs serve the same purpose, but force drivers to make it obvious that they are going to yield, even when there is no other traffic (Pharisee's law).
Finally, we can answer the question: "Are Jews still God's chosen people?".
No, they never were, so can't still be.
The Jews were chosen to carry the royal line and the Messiah (as described in the link at the end of Ken Graham's answer),
but that isn't what is meant by chosen people.
God chose Israel (not just Judah) to be his chosen people.
They were not chosen to be his favoured people, but to set an example to the world.
The pattern that the Pharisees noticed was not a coincidence.
God wanted the whole world to see what happens when people follow his laws and what happens when they disobey them.
As Paul notes in 1 Corinthians 10:11:
(KJV) Now all these things happened unto them for ensamples: and they are written for our admonition, upon whom the ends of the world are come.
(NLT) These things happened to them as examples for us. They were written down to warn us who live at the end of the age.
For the ancient Israelites, behaviour directly resulted in physical blessings and cursings.
For "spiritual Israel" (i.e. God's Church today, at the end of the age), behaviour results in spiritual blessings and curses.
And just as physical Israel was chosen to be a physical example to the world, God's people today are chosen (individually) to serve as spiritual examples to the world.
(KJV) There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus.
And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise.
(NLT) There is no longer Jew or Gentile, slave or free, male and female. For you are all one in Christ Jesus.
And now that you belong to Christ, you are the true children of Abraham. You are his heirs, and God’s promise to Abraham belongs to you.
As for the secondary question: "If the only path to the Father is through the Son, and Jews reject the Son, how do they still reach the Father? If they never accept Christ as their savior, how are they saved?", that really should have been asked as a separate submission.
The quick answer is that it is the same situation as for all the people throughout history that were born and died without ever being offered spiritual salvation.
That is the vast majority of mankind, and 1 Timothy 2:3–4 tells us:
(KJV) For this is good and acceptable in the sight of God our Saviour; Who will have all men to be saved, and to come unto the knowledge of the truth.
(NLT) This is good and pleases God our Savior, who wants everyone to be saved and to understand the truth.
God has chosen only a small flock of people to be saved during this current age, those that will rule with Christ in the Kingdom of God during the Millennium.
If God wants everyone to be saved, then those that were not offered salvation in this age will be given their opportunity in the Kingdom of God when they are part of the second resurrection at the end of the Millennium.
For more details, see my answer to What is the Order of the Resurrection of the Dead?.