Right now all this does not amount to much. However, if a district court and a court of appeals sides with the state, things will heat up considerably. Trump is riding the fence right now opposing the Alabama law but unless this is settled soon he's going have to take a position which will cost him votes among women voters either way.
A president should not even have the power to influence abortion laws anyway, they should be decided upon by each state, to suit the type of society their residents want there laws to reflect.
These recent laws are being written in a way to invoke a SCOTUS challenge. I think most of these laws were written much stricter than they would otherwise. They want the court to strike down Roe V Wade and send these things back to the states to decide. Then they will rewrite their abortion laws to be more inline with their voters, and not as strict as the are written now.
At least that is my thinking on this, from what I have heard from some of the politicians who wrote their new and very strict abortion laws.