The place where direct democracy works is as a veto. If the legislature passes a law and you collect X number of signatures against it, it goes up for popular vote. If the people don't approve, it doesn't pass.

Then you mitigate populism creating ill-conceived laws, because the legislature has to introduce and pass them first, but then the legislature can't as easily pass corrupt laws because they can be challenged and shot down by the people.

It would also be pretty great if you could challenge existing laws in this way, e.g. for any that hasn't been challenged this way in five years and you get the signatures again, it can be used for repeal.

We have unpopular laws that populism would like to see repealed - in the UK, the human rights act for example.

That is what constitutions are for. You can have things that require a supermajority to enact and therefore a supermajority to repeal.

If you do that, you get stuff like Brexit.