For example, steel is used by many businesses in the USA to create products which brings profit to those businesses. If the price of steel goes up due to a tariff, then it affects the profitability of those businesses. Businesses which actually produce steel itself would be able to obtain higher profit margins because they no longer need to compete with cheaper exports, but the overal affect on the economy is negative.
So if you look at the EU, yes, they should lower their tariffs, but they are hurting themselves with their tariffs. So the idea of reciprocol trade barrier reduction is incorrect. The country the eliminates tariffs benefits regardless of what the other country does.
But this does not happen for policital reasons and that is why the EU talks about trade wars. It's to look good to their voters. You'll notice in trade deals each country tries to get wins for targetted on certain special interest groups within their country. But this is in spite of the fact that if all trade barriers were removed it would result in the greatest possible benefit.
If you get tariffs, ie governmental inteference in trade, you can suddenly have large firms become unprofitable which leads to mass layoffs. So governments get up and say they blame the foreign government for it. And they may impose tariffs as a reprisal. In other words, if the actions of the foreign government are unpopular, it gives them cover to bring in tariffs that benefit certain special interest groups, increasing their political power within their country. And, in general, if a population thinks they are being attacked it usually galvanises them to support their leadership.