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You could do PR with the ballot of potential Reps distributed by district. When the election is settled the district Reps are assigned starting with the highest-skewed district. E.g.:

Overall vote: 60:40 (red:blue)

D1: 80:20
D2: 40:60
D3: 70:30
D4: 45:55
D5: 30:70

You can go randomly, round Robin, or winner-first to divvy up the districts, but essentially you would expect D1, D3, and D4 to be assigned their local red Rep (even though red "lost" in the close D4 race) and D2 & D5 to go blue

With more parties, random or round robin are a little more "fair" for the third party - winner first allocation could result in 3rd party getting the "whatever's left" district where they didn't actually get any votes.

It's not perfect, but neither is the current system.

your interpretation of FPTP is mostly correct however it’s a plurality that wins, even if it’s not 50%: if there are 3 candidates, you’d only the highest vote total out of all the candidates to win (which could be as low as 34%)

what you’re talking about though is representative vs proportional systems… in representative systems a group of people directly elects their representative (like in geographic districts, but it doesn’t necessarily have to be geographic: this can be seen in some cases where minorities are codified and those groups elect a minority representative), where in proportional systems your vote goes towards the government as a whole

i think this is far less of a black and white good vs bad than fptp vs stv/rcv/irv:

fptp voting counting leads to huge issues which force a 2 party system that will never represent the majority of people (through things like defensive voting, people vote less for the candidate they want and more for the candidate they think is most likely to win who isn’t the candidate they most don’t want), and recent american politics has shown that fptp also leads to much more polarising politics (in RCV systems candidates care about their 2nd, 3rd, 4th choice votes so they have to be as likeable as possible: they don’t want to come off as bullying they 3rd place candidate, because their voters really do matter)

proportional vs representative is more nuanced though… with representative systems you have someone who is there to represent your group specifically, rather a kind of often nebulous set of ideals… proportional meanwhile you do get more philosophically aligned candidates, but they always have to form coalitions with other parties (nobody has a majority: proportional governments are formed by lots of small parties/candidates) which means you can never really hold them to what they say: they’ll have to compromise a lot, and the government is very much sometimes beholden to the whims of marginal groups who hold the power (this has been happening a lot in europe at the moment where coalitions break down)

so in australia’s case we have a bit of a combination: for our house of representatives we use IRV/representative… we have districts, and we elect a representative, and those representatives form a government and the leader of the majority party is the prime minister. we also have our senate which is proportional (but still IRV), so they have a lot more small parties - including some far right shitbags

note though i am using RCV, STV, and IRV interchangeably but i believe they are different forms of RCV (and yes, i also believe RCV is both the category and a specific implementation). i think our ballot counting is IRV, but that’s based on some high school civics stuff so it may actually be another method and the teacher just said something generic

Without FPTP, the district result doesn’t matter at all, since it is the total number of votes that matter, not a designated winner of a district.

That depends entirely on what FPTP is replaced with. Any system with local representitives can be gerrymandered to reduce the representation of certain groups, with the exception of MMP where you can still gerrymander but it doesn't affect representation. That includes ranked choice, approval voting, etc. That's not to say these aren't better, of course with better local representation the effectiveness of gerrymandering is reduced, but it is not eliminated. The only way to eliminate gerrymandering is with a proportional system.

You can have ten Republican majority districts and one Democrat majority district. Then whatever voting system you have doesn't matter. To get rid of potential gerrymandering, you can treat the whole state as one district and have multiple winners.