That sounds even more British then our current system, which is really saying something.
Most of those "ideas" actually were used at some point or were at least considered. For example, more than a few Royal Mistresses received peerages (though at a time when peeresses couldn't sit), the planter class in one of the Caribbean islands, can't remember which one, owned a rotten borough to get themselves a voice in Parliament, etc.
The top two scoring candidates in the electoral college are then selected for a second round runoff vote. For this reason every major political party usually nominates two candidates for the primaries so there is chance that both candidates will be from their party. The "second place" candidate from each party is usually selected as their party's candidate for prime minister, since only members of the legislature can run. In the second round, a single non-transferable vote method is used, and the results of each state are again sent to the electoral college.
Do the parties expect that voters will have unusually strong party discipline? You say there are five parties. Imagine that the voting public splits 25, 20, 20, 20, 15 in favor of those parties. If each party puts up one candidate, one would expect that the voters would ordinarily send the guy from the 25% party and one of the 20% parties to the runoff. But if the votes are split between the two candidates put forward by the 25% party, both of them could easily finish behind two (or even more) 20% party candidates.
The reason that so many U.S. elections end up with two people from the same party in the final round is because either one party is so dominant that it can afford to split votes and take both runoff slots, or one party badly splits its vote and none of its candidates rise higher than 3rd place. With parties having no control over who can use their labels, parties have no central figure to rally behind and unite their base, making vote splitting even easier.