I'm not sure that allowing regexes would put an undue burden on the search engines. But if it ever becomes an issue, the search engine could easily deal with the problem by simply slowing down the search if it contains a regex.
I'd happily wait 2x, 5x, or even 10x as long for my query to complete if I could use a regex. For some important queries for which non-regex searches are inadequate, I'd even be willing to wait hours or days, since the alternative would be not being able to perform the search at all (or returning so many false positives as to be useless).