> Add a step immediately after parsing the regex, check if it is regular and can be handled by an engine like RE2, otherwise handle it using the normal regex engine.
This would be a bad idea. For one thing, it would be slower on non-pathological cases, since backtracking is faster in practice. But more importantly, devs who test on Chrome might unwittingly create regexes that would run exponentially slower in other browsers.
The real answer is that it's complicated. Backtrackers can certainly be faster in some cases, but not all. Moreover, as someone who has worked on regex engines for a while now, it's not clear to me how exactly to characterize performance differences (outside of pathological cases or cases that otherwise invoke catastrophic backtracking), so my guess is that it's probably mostly up to implementation quality and the optimizations that are implemented.
This would be a bad idea. For one thing, it would be slower on non-pathological cases, since backtracking is faster in practice. But more importantly, devs who test on Chrome might unwittingly create regexes that would run exponentially slower in other browsers.