Actually that doesn't work at all. In that sense you would lay justice in the hands of unqualified people, which is by far more devastating, and in several countries where the influence goes so far we can already see the terrible results of that.
Now it is a kind of misconception that judges cannot be held accountable. Even judges in supreme court can be held accountable, however the people judging them are normally not authorized to get into the case itself. Please note the perfect system cannot exist, however judges should be able to arise above politics. Having them chosen democratically kills the very purpose we have a justice system for in general, as a matter of fact, it would make their use completely obsolete. If there's a president having an election program that is a violation with every line written in the constitution but the people chose that idiot, they are also very likely to chose judges who would turn a blind eye to that or seek out the loopholes.
It is for a reason why the same Greeks who invented democracy were also in large numbers against it. Plato even advocated that democracy itself will be the biggest threat to democracy. His opinion will be proven right when judges are appointed by either politicians or the people. Some countries in Eastern Europe (some even EU members but not all of them) are already dying as democracies due to the court being entirely chosen by government officials, and quite a lot of them even revered by their people for that. A judge verdict is rarely what the people want to hear, but a judge must not be there to gain popularity. A judge must be there to uphold the law and constitution. The law is rarely convenient, but needed. All that chosen judges will lead to is the forming a banana republic, and unfortunately, that is a fact. I don't like that fact either, but we'll have to deal with it. Judges must be above public opinion... A chosen judge can never be that, as they do have election promises to keep. Judges must always be above politics in any way, and always be independent. You proposal makes conflicts of interest inevitable, unfortunately. I wish it were different.
Judges may be neither left, right, progressive, green, conservative, capitalist, communist, whatever. They must be like computers. Following their programming (which is coded in the programming language called "law") precisely. A judge must even be able to cast their personal opinion aside if it violates the law. C'est la vie