Why do Messianic Jews observe the Sabbath when First-Century Jewish followers of Yeshua/Jesus worshiped on Sunday?

Sadly, this is a historical misrepresentation. Despite what many "experts" claim, early Jewish believers did not change the day of rest. The day of rest was changed by Roman Emperor Constantine at the Council of Nicea in 325 C.E.

Although many publications claim Constantine was a Christian, the fact is that he worshiped "Sol Invictus", the Invincible Sun (in the sky). He made gentilized Christianity the Roman state religion solely to strengthen his political power base. To broaden its appeal he brought in many pagan customs.

However, this is not generally known in the modern gentile Church. Searching for Bible support, Christians base the claim that early Jewish believers worshiped on Sunday on two passages:

1Corinthians 16:1 Now about the collection for God's people: Do what I told the Galatian churches to do. [2]On the first day of every week,each one of you should set aside a sum of money in keeping with his income, saving it up, so that when I come no collections will have to be made.

That isn't talking about the day of rest, it is just financial advice put money away before you spend it.

Acts 20:7 On the first day of the week we came together to break bread. Paul spoke to the people and, because he intended to leave the next day, kept on talking until midnight.

This is a common Jewish custom called "chaverah", i.e., "fellowship". It starts Saturday night, after the sabbath is finished. Jewish days start at sundown, so the "first day of the week" runs from Saturday sundown to Sunday sundown.

But let us assume -- "for the sake of argument" -- that it was not a chaverah. The Bible states specifically that it was not normal circumstances -- Paul intended to leave the next day. The passage does not say they customarily met on the first day, only that they met to dine together.

In fact, the Bible does not say what was discussed. It could have been a simple planning meeting, or even just friends "catching up".

It is true that many gentile believers worshiped on Sunday, but Jewish believers generally did not change.