Like Brexit?
Big issues tend to be complicated issues on many levels. I don't see how you can agree that full-time politicians are required because people don't have the time or expertise to be involved in every aspect of the political process, but also that for big, important issues - almost certainly requiring the most time and expertise to fully understand the arguments and ramifications of all options - suddenly the same non-expert people are suddenly better than full-time politicians.
What happens is Brexit. A ridiculously complicated process (the UK leaving the EU) that could be done in all kinds of different ways (in the single-market or instead in the EEA, in/out the customs zone, free movement or not, a Norway-style semi-in arrangement, tariffs, regulations, membership of other EU institutions and so on) each with their own trade-offs and complications, on top of the ridiculous difficulty of simply trying to predict the known unknowns arising from unpicking a 50-year political and economic union.
https://www.politico.eu/article/uks-brexit-options-explained...
Instead we got:
"Should the United Kingdom remain a member of the European Union or leave the European Union?”
[ ] Remain a member of the European Union
[ ] Leave the European Union
where option two could mean anything, and in practise it let the Euro-hardliners and UKIP types define it as the hardest of hard Brexits and fuck the consequences. When it took years for Parliament to agree on the final result for what Brexit was (and the UK's system of government makes passing legislation easy ie. the govmt. has a majority in one house and can override the other house) it just shows how the idea that direct democracy fails at a certain scale. All nuance, complexity and context are stripped away and replaced by a single sentence asking a yes or no question that only shows people's feels over reals.
The fact that parliament became gridlocked for years afterwards as it failed to agree a sane way forward is not a great argument for avoiding referenda and relying on the wisdom of elected representatives.
So for the general case where we have representatives doing the voting, we should have a mechanism where we can switch representatives in cases where we don't agree on specific issues. And that's where principles from liquid democracy could bring solutions to the current coarse system that we have.
But I don't think there is a silver bullet.