I favor Approval, but am mindful of objections to its lack of preference.
It seems to me that the simplest extension of Approval is to add a second level of approval as in 3 slot methods. The ratings could be called Good, Ok, Reject as in 3-2-1 voting, but for now the rating names are not important.
I would also like the option of a 2nd round of voting if there isn't a clear majority.
With that in mind, I was wondering if this variant has every been considered:
3 slot method
Rating above Reject counts as Approval
If there is a beats-all (Condorcet) winner, that's the first round winner
If not, the candidate with the most top-ratings is the first round winner
If the first round winner has majority approval, the method is completed with no second round.
Otherwise, there is a second round, between the round-1 winner (R1W); the candidate who has highest total approval on ballots that don't approve of the first round winner (complementary approval winner = CAW); all candidates in the Smith Set; and the total approval winner.
In case of ties for beats-all or top-rated, all those tied candidates would be included in the second round.
In the second round, the method terminates with that round's R1W before the second-round step.
The intent here is to avoid strategic bullet voting, chicken dilemma or burial, while creating an incentive to add second choice approval for compromise candidates. Participation and FBC failures, if any, also appear to be minimal.
Is there an incentive for plurality top-rated candidate supporters to create a cycle to fall back to top-rated votes? Possibly, but I don't see any way that could happen without a second round and the additional risks that entails.
The reason I consider this as an Approval mod is that the optimal strategy is to vote all approved candidates as top-rated. But if you are concerned about a bullet-voting or chicken dilemma attack, there ought to be nothing lost by putting compromise candidates in second place.