I just learned at church the last few weeks some interesting things. I trust the pastor and believe that they are true.
One was that people think it was/is impossible for Jews to follow all 600+ Jewish laws. He said MANY observant Jews did/do. (Well, probably not their whole lives....) In fact, some of the leaders thought that if they could just get ALL the Jews to follow all the laws for just one day (but all on the same day!), that would bring the Messiah back!
And Jesus perhaps acknowledged this as well. Like when the rich man asked Jesus, “I have kept all the laws, what must I do to gain eternal life?” And Jesus replied something like “That is good that you keep all the laws, now go give away everything you own.”
And of course the Law does not equate to Sin. The Sermon on the Mount covers that! (One may keep the letter of the Law by not stealing, but then Jesus brings it down to what is in your heart and mind as well (to lust over a woman, car, cow, etc.)
And off topic - but I was surprised to learn that many people had the gift of healing back then. So that in of itself did not set Jesus apart too much. But nobody else could command nature (calming the storm, causing the fish to swim into the nets,, etc.) and of course raising the dead to life WAS a huge deal! And then like you said - the whole making Himself like God thing. ;)
Moses encountered this with the plagues. The priests could match many of his acts so the proof escalated.