I am not sure what Messianic Judaism means now anyway. Some within MJ reject the blessed Trinity. Some reject saint Paul's letters. How can one be sure what is what in a movement that's so broad?
Messianic Judaism basically consists of Jews who believe that Jesus is their Messiah, which grew to include Gentiles after Peter's vision in Acts 10. Generally it is made up of Jews who haven't brought into the lie that they have to give up their Jewish identity in order to follow their Jewish Messiah and Gentiles who recognize the inherent Jewish foundation of Christianity.
As with all organizations of sufficient size, you'll find a variety of beliefs. There are also Christians who aren't MJ's who deny the Trinity and reject Paul's letters, so that is not unique to MJ. I know there are some who argue that the Hebraic concept of God was different than the Greek concept, but I haven't looked that much into it. From what I recall, they understood that the Father is God, the Son is God, and the Holy Spirit is God, but that the Father is not the Son, the Son is not the Holy Spirit, and the Holy Spirit is not the Father, which amounted to a big shrug. The Greek mindset wanted to define everything and came up with the formalized concept of the Trinity, but trying to wrap our mind around that can also amount to a big shrug, so I don't get hung up over it.
I will say that if any of the other books in the NT contradict Jesus' words, such as in Matthew 5:17-19, then Jesus trumps them, but it doesn't have to come to that when Paul is correctly understand.
So as far as I can tell we have SDA, MJ, and Hebrew roots movements all pressing for people to observe the seventh day (Friday-sunset till Saturday-sunset) as Sabbath in compliance with one of the ten commandments. Some want dietary laws to be observed too and some want to keep the feasts mentioned in Torah too.
There are some MJs that hold that only Jewish Christians are required to keep the Mosaic law, but my understanding is that Jews would have considered anyone who taught against keeping the law to be in violation of Deuteronomy 13 and would have rejected them out of hand, if not tried to stone them. Many of them who were on the fence about whether to believe that Jesus was their Messiah were looking at whether Gentiles were coming to worship God in an acceptable way in great numbers. In their view, that would be a fulfillment of prophecy and an indication that Paul's ministry was from God and that Jesus was the Messiah. However, Gentiles showed no regard for the faith of Israel, then it was not of God and what Paul was saying was a lie.
I will say that those Christians who don't at least study the Feasts are missing out on some rich teachings about the Messiah.
I don't know what keeping all those things does for one's faith and spirituality. I don't know if any of the above movements teaches that keeping laws leads to salvation or not. Some say that grace is what saves and that commandment keeping has a role in cultivating holiness after salvation (after seems to be what they mean but maybe they teach something different from that, I am guessing to a degree on this matter).
I have not run across any who have taught that keeping the law leads to salvation, but that's not to say that aren't some people who are confused on the matter. As I've said before, the law is God's instructions for how to practice righteousness and 1 John 3:10 says it is the children of God who practice righteousness. It is an ongoing part of sanctification as we are made to be more like Jesus in how he thought and acted.