The FAA says "thou shalt not board" but honestly speaking, they don't have the time resources (and often the interest) so you can be zonked out on pills and board a jet and usually the flight attendants onboard are the first to notice. Trust me, they want passengers out of the gate area and onboard the jet so they can work the next flight.
ETOPS you're never too far to divert because you're always x amount of minutes from a diversionary airport and if dispatch wants you to divert to Keflavik instead of continuing on to Heathrow and you continue to Heathrow, it can get really sticky. Overweight landings honestly aren't that big of a deal in a medical emergency and we have checklists and guidance for those situations.
If something medical happens onboard a jet, at least at my airline, we have this "situation room". There'll be dispatch, a supervisor, meteorology, a fleet specialist, probably a mechanic, sometimes even a medical professional on duty, a duty pilot (basically like a pilot representative that's looking out for your interests when it comes to pilot stuff) and either a hotline to MedLink and probably some other people.
So when the call comes to divert all of the deliberations, considerations, performance considerations and medical logistics have already been discussed and, for lack of a better word... decided. The biggest role you have during a medical, if things go pear shaped is relaying information over voice to MedLink if the onboard facilities aren't working. Even flight control can refile you as "Lifeguard" without your concurrence.
I'm just the Uber driver during a medical. Target? Dennys? Strip Joint? Send me the performance figures and hope they have hotels because we're probably going to time-out.