Yes, I think that may be helpful to again review the statements regarding the covenant and the ten commandments.
The ten commandments were delivered in the context of the covenant promises:
Exo 19:3 while Moses went up to God. The LORD called to him out of the mountain, saying, “Thus you shall say to the house of Jacob, and tell the people of Israel:
Exo 19:4 ‘You yourselves have seen what I did to the Egyptians, and how I bore you on eagles' wings and brought you to myself.
Exo 19:5 Now therefore, if you will indeed obey my voice and keep my covenant, you shall be my treasured possession among all peoples, for all the earth is mine;
Exo 19:6 and you shall be to me a kingdom of priests and a holy nation.’ These are the words that you shall speak to the people of Israel.”
Exo 19:7 So Moses came and called the elders of the people and set before them all these words that the LORD had commanded him.
Exo 19:8 All the people answered together and said, “All that the LORD has spoken we will do.” And Moses reported the words of the people to the LORD.
The statement regarding bringing them to Himself, from Egypt, is showing His sovereignty over them, and His deeds done in kindness to them. He is their Redeemer, which is stated as a reason to enter into covenant, and serve Him.
We see this emphasized again when He speaks the commandments, and also in in Deuteronomy 5 in relation to the Sabbath.
God spoke the ten commandments from the mountain.
Exo 20:1 And God spoke all these words, saying,
Exo 20:2 “I am the LORD your God, who brought you out of the land of Egypt, out of the house of slavery.
Exo 20:3 “You shall have no other gods before me.
Exo 20:4 “You shall not make for yourself a carved image, or any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth.
Exo 20:5 You shall not bow down to them or serve them, for I the LORD your God am a jealous God, visiting the iniquity of the fathers on the children to the third and the fourth generation of those who hate me,
Exo 20:6 but showing steadfast love to thousands of those who love me and keep my commandments.
Exo 20:7 “You shall not take the name of the LORD your God in vain, for the LORD will not hold him guiltless who takes his name in vain.
Exo 20:8 “Remember the Sabbath day, to keep it holy.
Exo 20:9 Six days you shall labor, and do all your work,
Exo 20:10 but the seventh day is a Sabbath to the LORD your God. On it you shall not do any work, you, or your son, or your daughter, your male servant, or your female servant, or your livestock, or the sojourner who is within your gates.
Exo 20:11 For in six days the LORD made heaven and earth, the sea, and all that is in them, and rested on the seventh day. Therefore the LORD blessed the Sabbath day and made it holy.
Exo 20:12 “Honor your father and your mother, that your days may be long in the land that the LORD your God is giving you.
Exo 20:13 “You shall not murder.
Exo 20:14 “You shall not commit adultery.
Exo 20:15 “You shall not steal.
Exo 20:16 “You shall not bear false witness against your neighbor.
Exo 20:17 “You shall not covet your neighbor's house; you shall not covet your neighbor's wife, or his male servant, or his female servant, or his ox, or his donkey, or anything that is your neighbor's.”
Bringing the people out of slavery is again referenced as a claim upon them from their Sovereign. The Sabbath seems to play a similar role as the references to bringing from Egypt. He has legitimate rule over them, and more than just them, but over all things as Creator. God is not only Redeemer, but Creator, and they are to serve Him.
The people were afraid following hearing God's voice and asked Moses to not have God speak to them further, but for him to relay other commands.
Exo 20:18 Now when all the people saw the thunder and the flashes of lightning and the sound of the trumpet and the mountain smoking, the people were afraid and trembled, and they stood far off
Exo 20:19 and said to Moses, “You speak to us, and we will listen; but do not let God speak to us, lest we die.”
Exo 20:20 Moses said to the people, “Do not fear, for God has come to test you, that the fear of him may be before you, that you may not sin.”
Exo 20:21 The people stood far off, while Moses drew near to the thick darkness where God was.
God's appearance on the mountain, and speaking of the commandments was meant to test the people, and place fear upon them that they would not sin.
The sabbath command in the Deuteronomy 5 recounting of the ten commandments gives a different rationale, though not a new one as it was contained in the overall rationale for the covenant previously.
Deu 5:12 “‘Observe the Sabbath day, to keep it holy, as the LORD your God commanded you.
Deu 5:13 Six days you shall labor and do all your work,
Deu 5:14 but the seventh day is a Sabbath to the LORD your God. On it you shall not do any work, you or your son or your daughter or your male servant or your female servant, or your ox or your donkey or any of your livestock, or the sojourner who is within your gates, that your male servant and your female servant may rest as well as you.
Deu 5:15 You shall remember that you were a slave in the land of Egypt, and the LORD your God brought you out from there with a mighty hand and an outstretched arm. Therefore the LORD your God commanded you to keep the Sabbath day.
This again shows that the Sabbath command is tied to God's sovereignty over the people, and His claim on their lives due to being their redeemer.
The Sabbath is singled out as of utmost importance prior to the giving of the tablets of the testimony, of which they are part:
Exo 31:12 And the LORD said to Moses,
Exo 31:13 “You are to speak to the people of Israel and say, ‘Above all you shall keep my Sabbaths, for this is a sign between me and you throughout your generations, that you may know that I, the LORD, sanctify you.
Exo 31:14 You shall keep the Sabbath, because it is holy for you. Everyone who profanes it shall be put to death. Whoever does any work on it, that soul shall be cut off from among his people.
Exo 31:15 Six days shall work be done, but the seventh day is a Sabbath of solemn rest, holy to the LORD. Whoever does any work on the Sabbath day shall be put to death.
Exo 31:16 Therefore the people of Israel shall keep the Sabbath, observing the Sabbath throughout their generations, as a covenant forever.
Exo 31:17 It is a sign forever between me and the people of Israel that in six days the LORD made heaven and earth, and on the seventh day he rested and was refreshed.’”
Exo 31:18 And he gave to Moses, when he had finished speaking with him on Mount Sinai, the two tablets of the testimony, tablets of stone, written with the finger of God.
The Sabbath is shown to be prominent even among the ten as a sign that God sanctifies them, and that He is Creator. It is specifically said to be a sign with the house of Israel, and a covenant with them.
This seems to have some parallels to circumcision in the covenant with Abraham. It is not expressly called the sign of the covenant, but is singled out as an obligation above all for Israel as a covenant with the Lord.
In Ezekiel God again singles out the Sabbath as a sign in addition to the law by which one lives, alluding also to Ex. 31, when recounting His relationship to Israel.
Eze 20:10 So I led them out of the land of Egypt and brought them into the wilderness.
Eze 20:11 I gave them my statutes and made known to them my rules, by which, if a person does them, he shall live.
Eze 20:12 Moreover, I gave them my Sabbaths, as a sign between me and them, that they might know that I am the LORD who sanctifies them.
So based on all the above I would think the Sabbath is a sign given to Israel in the heart of the covenant document, and singled out among the spoken commands of God to Israel, as a continuing sign of God's sovereignty over His people as Creator, Redeemer, and the one who sanctifies them, or makes them holy.
If you have other insights please post them. If there is some agreement as to the purpose then we can discuss the other questions you raise about what happens if the Sabbath is removed from observance, etc.