On 10a the gemara says that Hash-m amde a pishara between yeshayahu and chizkiahu, no one wanted to go to the other, so Hash-m made chizkiahu sick.
How is this a pishara?If chizkiahu was walking, and hashgacha pruttisly saw yeshayahu, that would be a pishara. Why is it that the fact that chizkiahu was deathly ill a pishara?
The most straightforward answer is that neither the king of the entire nation, nor the leading prophet of the generation, would be taking leasurely strolls in the streets of the city, such that they would "bump into" each other. Even if they happened to be in the same vicinity, one going over to the other one would conflict which their specific Shitos. (You wrote that if Chizkiyahu was walking and he saw Yeshayahu, that would be a Pesharah. But Chizkiyahu certainly would not have gone over to Yeshayahu, because he maintained that the prophet should come to him !)
Therefore, Hash-m had to make a Pesharah such that one would be forced to go to the other, which is what happened. Chizkiyahu became deathly ill and Hash-m told Yeshayahu to visit him. (This raises a different question -- if Bikur Cholim is a Mitzvah d'Oraisa, then why did Yeshayahu not go to visit the ill king until Hash-m specifically told him to? See SIFSEI CHACHAMIM for an answer.)
Furthermore, it seems that the reason Hash-m made Chizkiyahu sick was not simply to bring Yeshayahu to him. Rather, Chizkiyahu was deserving of becoming sick and dying, as the Gemara goes on to explain, that it was part of the punishment for not performing the Mitzvah of Piryah v'Rivyah. Therefore, Hash-m, in His infinite wisdom, not only arranged that Chizkiyahu be punished for his sin, but that the very same act serve as the vehicle to get the prophet and the king together!
Moreover, perhaps we could suggest that Chizkiyahu's illness was part of the Pesharah in the following way. Chizkiyahu's claim that the prophet should come to him was that Eliyahu had come to Achav. Yeshayahu's claim that the king should come to him was that Yehoram went to see Elisha.
However, Chizkiyahu's proof is lacking, because in Eliyahu's case, Hash-m specifically told the prophet to go to the king. But He had not told Yeshayahu to come to Chizkiyahu. Yeshayahu's proof is also lacking, because the king Yehoram only went because he was in mortal danger, literally dying of thirst, but Chizkiyahu the king was not dying such that he should come to the prophet.
That is why the Gemara says specifically that Hash-m made Chizkiyahu ill and told Yeshayahu to go to him . Now, Chizkiyahu's proof was fulfilled -- the prophet had now indeed received orders from Hash-m to come to him! And that is why Chizkiyahu had to become ill -- so that Yeshayahu's proof would be fulfilled, that the king be deathly ill! Hence, a Pesharah.
(P.S. Are you related to the Hamer family in Denver?)