All verbs are written in kanji and kana combinations, but I am not able to find the reason for them to be written in this way.
This may not be the type of answer you expect, but the reasons is "because the Japanese government decided to do so".
The current government says all conjunctions (もし, そして, でも, ...), adverbs (とても, かなり, ...), particles (は, を, に, ...) and various auxiliaries (～だ, ～ている, ～た, ～ます, ～でしょう, ...) are to be written in hiragana. This rule was different in the old days. There were times when particles and auxiliaries were written in kanji, and there were times when particles and auxiliaries were written in katakana, and there were even times when the government seriously considered abandoning kanji altogether (like Koreans actually did). And as you know, verbs and adjectives have lots of conjugations, which are made possible by auxiliaries.
Practically, many Japanese people believe that mixing kanji and kana enables us to tell word boundaries easily without any spaces.