Hotspur would not claim the throne. He and Glyndwr were allied with Sir Edmund Mortimer, who was Hotspur's brother-in-law and married to Glyndwr's daughter. Sir Edmund had proclaimed his young nephew Edmund Mortimer rightful king, and that was the nominal cause of the rebels.
If the rebels killed both Henrys at Shrewsbury, then young Edmund would become king, with Hotspur or his father Northumberland as Lord Protector. Why should Hotspur attempt a risky usurpation when he could be the King's uncle and the effective ruler?
(Unless Henry's second son Thomas can rally the Lancastrians; but he was only 16. Henry's other sons John and Humphrey were 14 and 13.)
BTW, "Prince Hal" was only 17, while his supposed rival Hotspur was 39 - three years older than Henry IV.
Also, Prince Henry was wounded in the face by an arrow, so he could easily have been slain, and King Henry's standard bearer was cut down by Hotspur's ally Douglas, which suggests the King was in danger as well.