Within your arguments, there are a few instances of evangelical lingo and paradigm which I will illuminate as separate from scripture.
e.r.m. - If you are convinced that baptism is not a “work of righteousness”, you might want to do a little cross-referencing.
You know, of course, the directive Christ gave His disciples sometime before His ascension in Matt. 28:19-20 - the “Great Commission”. “Go ye therefore, and (#1) make disciples of all nations, (#2) baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: (#3) teaching them to observe all things whatsoever I have commanded you...”.
Baptism is included here along with these other activities in the list of tasks the disciples were to perform. “Works”, in other words. I also believe they are deliberately put in this particular order by design. Baptism is listed SECOND, because a person has to become a disciple / a Christian FIRST before they can make an overt, public confession of that discipleship by baptism.
I agree about the apostles baptizing disciples. We can get into the specifics of that at another time. Jesus instructed them to make disciples (learners) of all nations, to baptize those disciples, and to teach them to obey everything he had commanded them. It doesn't say they they were saved. In fact Jesus had told some of the Jews who had believed in himJohn 8:31-32,34-36 To the Jews who had believed him, Jesus said, “If you hold to my teaching, you are really my disciples.  Then you will know the truth, and the truth will set you free.”  Jesus replied, “Very truly I tell you, everyone who sins is a slave to sin.  Now a slave has no permanent place in the family, but a son belongs to it forever.  So if the Son sets you free, you will be free indeed.
Jesus put being set free (from sin) after being his disciples. So you don't have a case to say being baptized is done after being saved.
You then inserted "in other words" in order to add the idea of works, where it is not written. Evangelical influence number 1.
And you said before they can make an overt, public confession of that discipleship by baptism.
, which was never even the purpose for baptism in water in Jesus's name. No one in the Bible ever told anyone to get baptized as a public confession. Evangelical influence # 2. This comes exclusively from, and never prior to John Calvin“Baptism serves as our confession before men, in as much as it is a mark by which we openly declare that we wish to be ranked among the people of God, by which we testify that we concur with all Christians in the worship of one God, and in one religion; by which, in short, we publicly assert our faith…”
~ Institutes of the Christian Religion 4.15.13 John Calvin.
John Calvin was even wrong on the confession part. No one in the Bible told anyone to make a public "coming out" confession. Luke 12:4-5,8-9 "I say to you, My friends, do not be afraid of those who kill the body and after that have no more that they can do.  But I will warn you whom to fear: fear the One who, after He has killed, has authority to cast into hell; yes, I tell you, fear Him!  "And I say to you, everyone who confesses Me before men, the Son of Man will confess him also before the angels of God;  but he who denies Me before men will be denied before the angels of God.
refers to not shrinking back when asked, hence the denying part. And yes of course a disciple should preach God's word as well. But there was never expected a "Step", where someone comes out and says, "Hey world, I'm a Christian now!", much less for baptism to fill this non existent step. A Biblical baptism in water in Jesus's name can be done in the most private of rooms, in the most guarded castle with the baptizer and the baptizee and still be valid, because public confession was never the written purpose
But yes, they do need to be a disciple before their baptized in Jesus name for the Forgiveness of your sins, as is in Scripture.
Now go to Mark 16:20 and see the result of the apostles’ “Great Commission” of making disciples, baptizing, and teaching them. “And they went forth, and preached everywhere, the Lord *WORKING* WITH THEM, and confirming the word with signs following. Amen”.
By comparing this Matt. 28:19-20 text and its companion verse in Mark 16:20, baptism is proven to be a work.
God was working with them in much the same way He supported the Israelite army under Moses and Joshua. You're trying to use the reference of God working with them to include baptism in the definition of work. Basically grasping at straws. The evangelical playbook often tries to connect dots, in the absence of confirming scriptures. But sorry, if the authors viewed baptism in water in Jesus's name as a work, it would have been painfully explicitly stated, not hiding in obscurity, that you'd be forced to connect these kind of dots to try to see it.
Even Christ at His own baptism called it an occasion where He and John could both “fulfill all righteousness” (Matt. 3:15). Christ’s baptism was a work of righteousness.
Baptism is a physical rite having an administrator, a participant, the medium of a body of water, and is done in the name of God the Father, the Son, and the Holy Ghost, usually in the presence of certain witnesses. How could it not be a “work of righteousness”, especially since Christ gives it that description?
Are you familiar with the terms inferring and eisigesis? The burden of proof to authenticate a Biblical teaching is an explicit statement in the Bible. If these guesses that you're making were true, just like in your previous attempt to connect the dots, then someone in the Bible would have followed up with an explicit statement. You're trying so hard to see something that is not there, that you even added the word work to what Jesus said. Jesus didn't say it let's do this work of righteousness, Jesus said let us do this to fulfill all righteousness. Not to mention, Jesus was the only person in recorded human history who ever got baptized to fulfill all righteousness, everyone else who got baptized did so in relation to their sin. I get it, it's not just you. This is the warehouse of evangelical arguments that you draw from. It's the whole belief system. I've heard these arguments time and again. But just like they try to turn Luke 18:13-14 into a possible beginning for "the sinner's prayer" and there was never any follow-up on this from anyone, revealing that the story was only a lesson on humility, so the Matthew 3:15 attempt has similarly failed to be confirmed by anyone calling baptism a work, which I believe would have been done if they actually considered it that way. There must be an explicitly stated confirming scripture somewhere for that idea to be true, and there's not. Inference is not enough.