I believe what the video says about finding definitions of symbols from within scripture. However, the leap to say that observing the wrong day as Sabbath is the Mark of the beast doesn't work for me for quite a few reasons. Here are just 2 :
1. The Sabbath is a shadow of Christ.
Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days:
Which are a shadow of things to come; but the body is of Christ.
Colossians 2:16-17
{Note that "days" is not found in the underlying Greek}
One man esteemeth one day above another: another esteemeth every day alike. Let every man be fully persuaded in his own mind.
He that regardeth the day, regardeth it unto the Lord; and he that regardeth not the day, to the Lord he doth not regard it. He that eateth, eateth to the Lord, for he giveth God thanks; and he that eateth not, to the Lord he eateth not, and giveth God thanks.
Romans 14:5-6
For unto us was the gospel preached, as well as unto them: but the word preached did not profit them, not being mixed with faith in them that heard it.
For we which have believed do enter into rest, as he said, As I have sworn in my wrath, if they shall enter into my rest: although the works were finished from the foundation of the world.
For he spake in a certain place of the seventh day on this wise, And God did rest the seventh day from all his works.
And in this place again, If they shall enter into my rest.
Seeing therefore it remaineth that some must enter therein, and they to whom it was first preached entered not in because of unbelief:
Again, he limiteth a certain day, saying in David, To day, after so long a time; as it is said, To day if ye will hear his voice, harden not your hearts.
For if Jesus had given them rest, then would he not afterward have spoken of another day.
There remaineth therefore a rest to the people of God.
For he that is entered into his rest, he also hath ceased from his own works, as God did from his.
Let us labour therefore to enter into that rest, lest any man fall after the same example of unbelief.
Hebrews 4:2-11
And if you counter by speaking of the need to observe the sabbath, you are hearkening back to the shadows - not to the reality (again...see Col 2:17). Christ fulfilled the Sabbath. If we are in Him - if we believe in Him - it is fulfilled and we are not bound to observances. That is, the fulfillment in Christ supercedes the establishment of an ordinance. The one who is in Christ will naturally rest, but the one who is treating the Sabbath as a commandment only is showing that they are making their boast in the Law. Those who are ruled by "have to" are merely observing the Law and are not fulfilling the Sabbath.
2. The Sabbath was a sign between God and the children of Israel.
Speak thou also unto the children of Israel, saying, Verily my sabbaths ye shall keep: for it is a sign between me and you throughout your generations; that ye may know that I am the Lord that doth sanctify you.
Ye shall keep the sabbath therefore; for it is holy unto you: every one that defileth it shall surely be put to death: for whosoever doeth any work therein, that soul shall be cut off from among his people.
Exodus 31:13-14
And I gave them my statutes, and shewed them my judgments, which if a man do, he shall even live in them.
Moreover also I gave them my sabbaths, to be a sign between me and them, that they might know that I am the Lord that sanctify them.
But the house of Israel rebelled against me in the wilderness: they walked not in my statutes, and they despised my judgments, which if a man do, he shall even live in them; and my sabbaths they greatly polluted: then I said, I would pour out my fury upon them in the wilderness, to consume them.
Ezekiel 20:11-13
I am the Lord your God; walk in my statutes, and keep my judgments, and do them;
And hallow my sabbaths; and they shall be a sign between me and you, that ye may know that I am the Lord your God.
Notwithstanding the children rebelled against me: they walked not in my statutes, neither kept my judgments to do them, which if a man do, he shall even live in them; they polluted my sabbaths: then I said, I would pour out my fury upon them, to accomplish my anger against them in the wilderness.
Ezekiel 20:19-21
And the argument that this is a sign between all the people of God and God is easily answered in that the people of God in the time of the New Covenant observed the Lord's Day - the day upon which Christ was resurrected. And did the Jerusalem council lay upon the people of God the necessity to obey the sabbath of the children of Israel? No :
But there rose up certain of the sect of the Pharisees which believed, saying, That it was needful to circumcise them, and to command them to keep the law of Moses.
Acts 15:5
So this wasn't just about ceremonial law. It was about all the law that came through Moses.
Now therefore why tempt ye God, to put a yoke upon the neck of the disciples, which neither our fathers nor we were able to bear?
But we believe that through the grace of the Lord Jesus Christ we shall be saved, even as they.
Acts 15:10-11
The Law was a schoolmaster to bring men to Christ. Not the way a man would be saved.
Simeon hath declared how God at the first did visit the Gentiles, to take out of them a people for his name.
And to this agree the words of the prophets; as it is written,
After this I will return, and will build again the tabernacle of David, which is fallen down; and I will build again the ruins thereof, and I will set it up:
That the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who doeth all these things.
Known unto God are all his works from the beginning of the world.
Wherefore my sentence is, that we trouble not them, which from among the Gentiles are turned to God:
But that we write unto them, that they abstain from pollutions of idols, and from fornication, and from things strangled, and from blood.
For Moses of old time hath in every city them that preach him, being read in the synagogues every sabbath day.
Acts 15:14-21
The synagogues were the Jewish places of worship and the sabbath was (and is) the seventh day of the week. The law of Moses included such things as absttaining from idols, fornication, things strangled and blood. That was in the law. So it wasn't like the Jerusalem council was adding to Moses - but rather saying that those that were not of Israel need not be weighed down (as they said, even they couldn't bear it) by the law of Moses. And note, too, that the ten commandments include idolatry and fornication. This is not to say that the moral law is not universally applicable (it is!) - but rather that there was no need to make of it a necessity for salvation. Returning to verse 5, this was in response to the Judaizers. And since they mentioned the sabbath explicitly (as did Paul), we can contrast that with "The Lord's Day" which is the first day of the week when Christ arose.
And upon the first day of the week, when the disciples came together to break bread, Paul preached unto them, ready to depart on the morrow; and continued his speech until midnight.
Acts 20:7
Now concerning the collection for the saints, as I have given order to the churches of Galatia, even so do ye.
Upon the first day of the week let every one of you lay by him in store, as God hath prospered him, that there be no gatherings when I come.
I Corinthians 16:1-2
I was in the Spirit on the Lord's day, and heard behind me a great voice, as of a trumpet,
Revelation 1:10
Were that the sabbath, you can be fairly certain John would have called it as such since he knew very well exactly what the sabbath was and when it was to be observed :
The Jews therefore, because it was the preparation, that the bodies should not remain upon the cross on the sabbath day, (for that sabbath day was an high day,) besought Pilate that their legs might be broken, and that they might be taken away.
John 19:31
And even if he were speaking of the seventh day, his emphasis on "the Lord's Day" would be to remove the Jewish implications of it.
There are other reasons to reject Sunday observance as the Mark of the Beast, but since it is clear that specific days are not bound to the believer's conscience, I think these two are the most pertinent.