> abolishing tariffs
Like by having a single market and customs union? Many FTA. Abolishing tariffs is what the EU does. 1 standard, no tariff, less government.
>abolishing immigration restrictions
Free movement of labour is one of the four freedoms of the EU. The main reason Teresa May is citing for leaving the single market is so she can increase immigration restrictions
The EU does not increase government. It sets common standards that mean trade is much easier, and encourage the rest of the world to abide by those common standards and to enjoy the same freedoms. Instead of every country having customs rules with each other, a country outside the EU can make a deal with one authority and then trade with all of the countries.
Chile is a good example. I imported from Chile in my last job. It was a doddle, because Chile have a FT deal. I apply Common Customs Tariffs of 0. Before the EU a small South American producer would have had to have worked out a deal with each of the 27 states, each with their own standards.
If you want to license medicines in the EU, one process covers the whole EU. That is less government! If you want to sell electronics, again the CE mark covers all of Europe and many other countries. If you match EU standards in any area you get equivalent treatment to a local producer. That is less government.
The UK is working out how it is going to replace all of those agencies with new ones. That is more government doing the exact same thing.
On the contrary, all I hear about on the news about the US trade is about Trump and his trade war with China (this time). When he tried the same with the EU, he was dealing with a much larger market for US goods, and the EU took the sting out of it. Lower tariffs and less government.
I cannot for the life of me see how having more independent nation states with their own trade deals will create anything other than more different tariffs, protectionist regimes and more government.