The issue in your example is of course that both parties send all their candidates to the election, instead of just their most popular one, and that dilutes the vote, but the same would happen if they did that in FPTP.
It's only in systems like approval voting or instant run-off that parties can afford to send all their candidates.
In practice, most US elections are preceded by explicitly partisan primaries, and that's the status quo you should be comparing against.
So unless you actively ban any kind of candidate selection by the parties, that is what would also happen in the US.
Well, that is just plain two round system, it is pretty common and definitely better than FPTP. The bizzare outcome you describe may also happen with FPTP, you just need one candidate from party A and several candidates from party B. The solution is same as in FPTP system - each party endorses just one candidate.
I would differentiate it from a two-round system in that voting takes place on a single ballot paper.
In two-round elections there is the ability to use the information about who is in the top 1 to cast the second vote.
In Supplemental Voting an elector has no information as to who's going to be in the top 2 when they cast the second preference.
https://ballotpedia.org/Washington_Treasurer_election,_2016
Duane Davidson (R) won against Michael Waite (R) after 3 democrats split the vote in the primary:
Republican Duane Davidson 25.09% 322,374
Republican Michael Waite 23.33% 299,766
Democratic Marko Liias 20.36% 261,633
Democratic John Paul Comerford 17.97% 230,904
Democratic Alec Fisken 13.24% 170,117If outcomes in general are better, I'll take the risk that they're occasionally worse.
Basically, let's say party A is unpopular, and has 2 candidates. Party B is extremely popular and has 7. Party A's candidates each get 15% of the vote, Party B's candidates each get 10%. Even though most voters voted for Party B, they're now stuck in a runoff with two candidates from the opposite party.
Some people argue that this forces parties to be disciplined and force out candidates they deem "unsensible" or "extreme". But in practice this often doesn't work, and even if it did I feel like the mark of a good voting system is that voters feel safe expressing their preferences without having to be strategic.