As far as MIs, though, quite possibly the greatest factor affecting your survival is how quickly you can get carted away to the cath lab, and I can't see how continuously monitoring anything is going to get you there any sooner than just paying attention to that crushing substernal pressure radiating to your left arm and calling 911.
(2) Even EKG changes aren't instantaneous. You'll have been having chest pain for quite some time before you start showing hyperacute T waves.
(3) But, finally, you have to target your measurements. Standard continuous cardiorespiratory monitoring is probably going to show you an increased heart rate and increased respiratory rate. Not very helpful. Continuous cardiac monitoring doesn't have the same resolution as an EKG; you're only probably going to get useful data if the patient is actually wearing a the full-12 lead set of electrodes, and again, see (2). And continuous troponin measurements would be extraordinarily low yield. I can't imagine what it would gain you over the usual q6h measurement, nor is there a point of measuring it when you're not symptomatic.
I can't think of anything we could've measured on Natasha Richardson that would involve a monitoring machine that she could've worn. Maybe if there were such a thing as a wearable CAT scanner, but would the radiation exposure be worth it? Probably the only thing helpful would've been continuous neuro checks, and you need a human being to do those.
No problem is so large it can't be fit in somewhere.