Conversely, while we could control immigration, business and agriculture depend on cheap foreign labour, so we would be crippling our economy by doing so...reference the ferocious backpedaling by dozens of leading Leave supporters shortly after their Pyrrhic victory.
We may not have to contribute to the EU budget (depending on our future EU relationship), but the loss of Single Market access may mean any money saved disappears into the aether that is economic recession.
Parliament may become more sovereign, but that depends on your definition of sovereign - after all, in this whole Brexit debate, Parliament's role has been rather ignored. If anything, Parliament's sovereignty has been undermined by the referendum. Also, we make deals with foreign entities all the time that restrain what Parliament can and can't do regularly - witness WTO and WHO rules and regulations, for example.
Regarding regulation, many of the EU's regulations will continue in existence but on a UK level, as oftentimes it is the UK that demands such high standards from its European neighbours and gets them enacted in Brussels. Outside the EU, we will be obliged to continue with them in order to keep our products and services compatible with the trade arrangements of our nearest and greatest trading partner. Furthermore, many of the EU-level regulations served to ensure a level playing ground for the entire EU, meaning in practice that regulatory burden fell - better one big enormous document to fill in than 28 of them, after all.
As for the speed of signing trade deals, there's no real evidence that the EU is any slower than any other country in terms of making them. We'll have to see, but smaller countries tend to get messed around with more by larger ones, so an advantage of EU membership will be cut away.