Approval Voting is robustly superior to Borda (and essentially all ranked methods), given you have any strategic voting. Here’s a graph of Bayesian regret values from the William Poundstone book Gaming the Vote.
A potential improvement to Approval Voting would be Score Voting (Approval Voting is effectively Score Voting on a 0–1 scale). There’s some evidence that STAR Voting (Score Voting followed by an automatic/instant top two majority vote) is even better by being more robust to strategy.
If there’s one takeaway from Gaming the Vote, it’s that you want to avoid ranked voting methods in favor of rated voting methods, like Score Voting and Approval Voting. This is counterintuitive but the data robustly supports it. Changing from Approval to anything resembling Borda would be a huge mistake.
BOSCore’s analysis uses Mathematician Donald Saari’s interpretation of Kenneth Arrow’s impossibility theorem to determine that the Borda count voting system can be considered a “fair” method of voting
Donald Saari has made some rather egregious mistakes in his analysis, as is explained here by a Princeton math PhD named Warren Smith, who is arguably the world’s top expert on voting methods.
Simply put, it is utterly false to claim that any ordinal (ranked) voting method can satisfy Arrow’s criteria. For that you need a cardinal (rated) method like Score Voting or Approval Voting. Moreover, Arrow’s criteria are not even particularly important. What we care about is the performance of a voting method, not arbitrary criteria. Kind of like how you want a race car that gets the best average race time, not necessarily the race car with the lowest drag or the most horsepower. The measure of “race time” for a voting method is Bayesian regret (alternative expressed as voter satisfaction efficiency).
So if your conclusion to use a Borda-like system is based on anything Donald Saari said, you are essentially getting everything wrong. Saari is considered by most voting methods experts to be a bit of a fringe eccentric.
Borda is famous for having exclaimed, “My scheme is intended only for honest men” (quoted in Black 1958, p. 182), when the susceptibility of his rule to strategic manipulation was pointed out.