Operator size, the markets you are in or weighing, the situation you want to discuss, and an indication of timing. That is enough for me to come back with a useful first response. The structure that works: one or two sentences on the operator (regulated or sweepstakes, approximate size measured in GGR or FTDs per month, stage); one or two sentences on the situation (market entry, repositioning, exit preparation, specific tactical question); one sentence on timing (when you need to make decisions, what the window looks like).
With that information, the same-day reply is genuinely useful: a first read on whether the situation is something I can help with, what shape that help would look like, and what the right next step is (discovery call, diagnostic engagement, or pointing you to a different consultant if I am not the fit). Without that information, the reply has to be a generic "tell me more", which wastes both sides time. The opening message that works best is the one that is concrete and specific.
The opening message that works worst is the one that opens with a pitch on the operator without saying what the actual question is.