It's currently illegal. Under current election law, districts have to only have one representative.
There was a period where states were using at-large districts where everyone in the state could vote for every Representative. So if California used that for all its Representatives, each person could vote for fifty-three candidates. If 50% plus one voters all picked the same fifty-three candidates, their picks would get all the seats. So in current California politics they'd go from mostly Democrats with some Republicans to all Democrats.
At the time that this was a problem, Southern states were using them so that African-American voters would get no candidates of their choosing. Court cases and other hijinks ensue. Currently single candidate districts are mandated in federal election law. So that law would have to change so as to allow states to choose proportional representation in multi-candidate districts.
The current system disadvantages third parties, which helps the main two parties. This leaves the two main parties unwilling to support this kind of reform. Even if the Democrats make small gains relative to the Republicans, they could still end up losing overall. And the Republicans could lose twice: once to the Democrats and a second time to third parties.
For that matter, in some states, it would be Democrats who would lose twice. Massachusetts and Connecticut have all Democratic delegations. Illinois and Maryland have Democratic gerrymanders.
Democrats are committed to redistricting reform by commissions. They believe that nominally non-partisan commissions are easier to promote. They also don't have the same concerns about losses to third parties with commissions.
Party list v. Single Transferable Vote
Party list gives more political power to parties. Instead of voting for candidates, people vote for parties. If a party wins nine seats, nine of its candidates are picked off a list by party leadership. To the good, each party is awarded a number of seats proportionate to its share of the vote. To the bad, people lose the ability to vote for individual candidates. That choice is made by party leaders instead.
Things don't have to work that way. Single Transferable Vote (STV) allows voters to choose their lists. So voters could choose lists that includes multiple parties if they want. Or choose a straight ticket.
Note that when I say STV, I mean the class of methods where a voter lists multiple candidates but only gets one vote. There's also a specific method of working out which candidate gets a particular voter's vote that is called STV. But I don't want to limit to just that one. Any of that class would be suitable.
Party List is easy for voters (just pick a party) and easy to count. About as simple as plurality voting. STV is more complicated. Voters have to prepare a whole list and order it. Counters have to read in every entry on the list and consolidate with other voters.
In any case, people opposing proportional representation can paint it with any of the problems from either solution.
Proportional representation groups together multiple districts. For some, the idea of local representation is important. It's still possible to cast your vote for the local candidate with STV, but with geographic districts everyone else is limited to the candidates in the local district. So the most local candidate who wins may be more local.
One of the election reforms created something called a majority minority district. Majority minority districts were a reaction to states spreading out their African-American voters so that there were always more white voters in their voting districts. By mandating that redistricters had to include majority minority districts where possible, the system encouraged districts that African-American candidates could win.
The normal proportional representation system does not have majority minority districts. Instead, all districts are lumped together. That said, there is no reason why a state couldn't be divided into majority and minority seats. In fact, states could have more minority seats under a proportional system, as they aren't limited to just grouping people who live near each other.
The normal argument is that in a proportional system, special handling for minorities is not necessary. If people want to band together to support candidates who have something in common with them, they can. So if women wanted to vote for only other women, they could. If a minority is too small to support a full seat, then that won't help. But the current system can't help in that case either.
It's mostly a combination of institutional inertia and negative perceptions. The negative perceptions reinforcing the institutional inertia. While there are benefits to reform, it's not evident to everyone that the benefits outweigh the challenges.