I've heard various proposals.
- From Republican Newt Gingrich in the 1990s: reform the Supreme Court such that it could not overturn a federal law without congressional approval. I.e. Congress could negate a Supreme Court decision by voting against it, in effect making Congress a more supreme court.
- From Democrats recently: add enough new seats to the Supreme Court to offset the existing partisan balance and swing it to them.
- From Democrat Franklin D. Roosevelt in the 1930s: add one seat to the Supreme Court for each justice over seventy and a half. At the time, that would have meant six seats.
A common law court system from the United Kingdom is different from Italy's civil law system. But the UK and Italy are also both parliamentary systems. In effect, the House of Commons in the UK is the government and only makes decisions by majority rule. This is in contrast to the US, which has multiple branches and various super majority requirements. The US is specifically designed that the executive, legislative, and judicial branches check and balance each other. In Italy and the UK, the law can be changed by the legislature if they don't like a court's interpretation. In the US, the constitution can only be changed with a higher consensus (specifically three quarters of the state legislatures must agree). The Supreme Court is more powerful in the US than equivalent courts in the UK and Italy. As such, it requires more protection from undue influence.
Gingrich's proposal would have essentially made the US more like the UK and Italy. The legislature would be able to overturn a top court decision with a majority vote. As is, this requires a constitutional amendment.
Incidentally, the modern era with every justice being a partisan battle started in 2005. The last two justices before that were voted in overwhelmingly. Even in 2005, John Roberts received seventy-eight votes. It was Samuel Alito who was controversial. Antonin Scalia is not mentioned because he was overwhelmingly confirmed per se. The real reason to point to Scalia is that he was a known conservative and overwhelmingly confirmed. Anthony Kennedy received a unanimous vote after Scalia. Stephen Breyer received eighty-seven votes in 1994.
Why other systems wouldn't work in the US
Part of the problem here is that the US legal system is dominated by Democrats. While there are certainly individuals who are conservative or Republican, the overall system is under Democratic control. The kinds of judges who would be nominated by an independent panel or by the Bar Association would be more liberal in their jurisprudence than the general populace or the politicians that they have elected would prefer. There is a specific example of this in Iowa, where judges are selected by such an "independent" panel. And in the 2010 retention election, the three who were up for retention all lost.
The US system has traditionally favored compromise. Scalia may have been conservative, but his jurisprudence on first amendment (e.g. flag burning) and fourth amendment cases was more liberal. Kennedy, Sandra Day O'Connor, and Byron White were the median voters on many cases, siding with different partisan sides on different cases. There is less compromise needed if the judiciary, the legislature, and the executive each pick their own candidates without having to adjust for the others.
A short term, like nine years, would allow for an administration to replace most of the court. In the US, presidential terms are four years, but most serve for eight. Further, the other branches tend to move with the executive. From 2009 to 2016, Obama appointed over three hundred judges. So his choices would tip the selections from the courts. From 2007 to 2010, the Democrats controlled the legislature. So four years of Democratic picks, replacing half the legislative judges. So Obama would have essentially had a court of candidates picked either by himself or by his supporters for the next five years.
If we ignore the institutional preference for liberals, that would have been even worse in the other direction during the George W. Bush administration (2001-2008). Congress was Republican from 1994 to 2006 with a brief period with a Democratic Senate in 2001-2002. All the legislative picks would have been Republican by 2003 And stayed that way until 2007. Two thirds of the presidential picks would have been Republican in 2006. The judicial picks would have swung to being Republican by that time. In 2007 and 2008, Democratic legislative picks would be offset by Republican presidential picks. And the judicial picks would have continued to swing Republican.
To put this more concretely: two thirds of five (each branch selects five in the Italian system) is ten thirds, which is between three and four. That means that it is possible that in 2006 with a Republican Congress and president, that nine of ten executive and legislative picks would have been Republican appointees. Chances are that the judicial appointees would have been swinging that way too, but even eight of fifteen is already enough for a majority. Republicans would have had control of the court. As is, in 2006, the consensus pick Kennedy was sometimes siding with liberals on partisan issues, restricting the Republicans.
Similarly, in the Obama situation, let's assume that the legislature picked three candidates and Obama two by the end of 2010. Starting in 2011, the legislature would be back to picking Republicans. But they would be replacing Republicans. There'd still be two Democrats. By the end of 2016, Obama would have had at least four picks and possibly five. So six or seven. And if we assume that the most recent three judicial picks were chosen by the Obama appointees in the judiciary, that's nine or ten. Contrast with actual system, where consensus pick Kennedy sometimes sides with conservatives on partisan issues, restricting the Democrats.
You can try tinkering with terms to fix that, but consider that from 1932 to 1952, Democrats controlled the presidency and most of that time controlled Congress as well. From 1968 to 1992, Republicans controlled the presidency for all but four years.
The real problem in the US at the moment is that Supreme Court decisions are consequential and final. There is no practical way to reverse a Supreme Court decision except another Supreme Court decision. And politics only makes this worse.
Anyone could have predicted that when Harry Reid ended filibusters for most nominees that Republicans would extend that to Supreme Court nominees. And the Democrats fell into that trap with opposition to Neil Gorsuch. Now think about what would have happened with the Kavanaugh nomination if the rule still existed. Some Republicans, like Jeff Flake, Bob Corker, Susan Collins, and Lisa Murkowski would be reluctant to end the rule over an actual controversial nominee. Kavanaugh would be replaced with a more consensus choice. But it was more important to get back at Republicans for not confirming Merrick Garland than to keep the rule.
In the UK, any law can be amended with a simple majority vote. Even if the law itself calls for a super-majority, they can simply change the law. So if the court rules against parliament, they just change the law that the court used.
In a civil law country, there is no implication by precedent as in a common law country. This makes it difficult for courts to extend rights that aren't specifically in the law. And like in the UK, they can generally add a new law if the court fails to extend a right that they believe should exist. Or another way of saying this is that in a civil law country, every judge is a textualist. In the US, that may be a radically conservative position, but in a civil law country it is the law. If the law is wrong, the legislature is expected to change it.
In the US, courts are supposed to check and balance the powers of the executive and legislative branches. The government is intended to only act in ways supported by laws passed with a constitutional basis. When it overextends, the courts are supposed to rein it in. But does anyone really expect a justice appointed by a president to rein in that president? For that reason, most of the justices are holdovers. And the courts are more powerful than in other countries where their role is more to help the legislature and less to check it.
The truth is that "independent" commissions tend to lean liberal in the US. They engage in actions opposed by many voters. This makes it easy to rail against decision made by unelected judges. In other countries, this matters less. Changes in law undo the decisions. But the US has many provisions that are not subject to change by a simple majority. This makes having judges who are representative more important.
If someone was looking for a change to make, it might be better to concentrate on how the constitution works rather than how judges are selected or how long they serve.