"Счастливый случай" is more of an equivalent of “happy accident”saying in english. It is an almost literal translation of the saying that just happens to work the exact same way in both languages.
That doesn’t mean that “happy” (in english) is synonymous with “lucky” or that it makes sense to apply “happy” the same way outside of that saying in english. Same thing in russian with "счастливый случай". If you use "счастливый” to apply to something outside of that saying, it will mean “happy” 100% of the time. If you try using it in place of “lucky” in russian, I can guarantee that people will be very confused even after you explain your reasoning for doing so.
P.S. “случай” literally means accident/happenstance, it doesn’t mean chance. There is a word for chance in russian that is extremely commonly used, and it is pretty much the same as the english word (“шанс” pronounced as “shahns”).
"счастливый случай" (lucky chance), guy probably meant