First of all, great idea! It could totally work.
@Povilas raised a good point in his contribution. If you keep the alarm at home, it may drive the other occupants crazy since all of you do not get up at the same time. If you take the alarm with you, you disturb other people who are there for some peacetime. There are jogging parks where people meditate, perform yoga, etc. which need some quiet. A person running with their alarms ringing will be frowned upon.
Alternatively, what if the same input (increased heart rate) that turns off the alarm is used to unlock the phone? This eliminates the problem of keeping the alarm at home and not knowing whether it is turned off. It also does not disturb the other people on the road, parks, etc.
If you are worried about emergencies, the user, before running, could be allowed to make calls or send an emergency message (this usually has a hotkey) to their specified contacts. All the other apps (social media, games, news, etc.) remain locked until you achieve the threshold heart rate.
Another advantage of this is you stay away from social media until you take care of your health (this sends a good message; something similar can be the tagline of the app :) ). Also, it does not need additional hardware.
The user cannot simply wake up and start running. They will have to perform a few activities like brushing their teeth, using the washroom, having a glass of water or some food, change clothes, wear shoes, which will take 15-20 minutes. Also, when the user starts running, it will take another 5 minutes to achieve the desired heart rate. The alarm will be continuously on for at least 20 minutes, which will drain the battery, too.