A few times I have argued with people about whether G-d really cares about all the details Judaism is obsessed with. About how exactly we wash our hands, which shoe to put on first, in which order to say the blessings, and so on. I mean it’s fun to argue these things (ok, I’m a nerd) and it creates a tight community. But try to explain to an outsider that you need to put on the right shoe first (Kitzur Shulchan Aruch 3:4) — he’ll think you are crazy.
That you wash your hands before eating is certainly a good idea and I get that before the advent of modern science you make a religious requirement out of it to make sure people do it. And I also get that people develop traditions about small details. And I know how to wash my hands and I do it that way on Shabat because it’s part of the experience, sort of. But I don’t honestly think something bad will happen if I wash in a different way. I don’t see an important difference and I don’t think G-d does.
You may say that maybe in old times people argued the same way about hand-washing itself. They didn’t understand hygiene and may have said "well, I don’t see a difference in whether I wash my hands before eating or not" — well, yeah. We humans are limited in our understanding. I cannot prove that the order of hands makes no difference, but neither can anyone prove that it does. All we can do is read up on the history of the tradition and the arguments brought in the discussion. And make an informed decision.
My personal opinion is that G-d would appreciate it more if we spend our energy on the stuff that makes the world a better place, like helping people, treating them with respect, gossiping less, and taking care of the environment. The rest is sugar coating which is important for creating an identity as a group, but not absolutely essential for a relation with G-d or being a good person.