Restricted Sprint?

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I don't think so. Not only is encumbrance and fatigue one of the most annoying features in games (it has absolutely no place in games - if I want to feel encumbered I'll go run a marathon while carrying a washing machine), it makes literally no sense within the horror idiom. The human body is capable of almost anything in times of intense fear - if you're running away from a monster you simply are not going to stop sprinting and suddenly start walking because of a want for energy. It's possible to completely wear out your muscles (to the point where you will be permanently injured) but the body certainly prefers that to dying - so you keep going for as long as physically possible.

I agree, most of the time it's more annoying than immersive. It might be interesting though, I think, to have a similar approach for encumberance as we already have for the insanity mechanic:

I read in an interview, that when it was first implemented, going insane had real gameplay repercussions (like decreased walking speed or something like that). However, Frictional felt that this was more annoying than anything else, so they reduced it to an almost purely atmospheric feature.

I think this could also work for fatigue. Like, say, you're running away and you start hearing your characters heartbeat become faster and louder at some point, you hear his breath become ragged or maybe other sounds get muffled and you get that ringing in your ears you sometimes get when you're close to collapsing. The point being to make the player feel like he won't be able to go on for much longer without actually enforcing it. It could add another layer of tension to chase sequences without being frustrating and risking more frequent player death.