Tab Search: Remove clear search icon from input field

This CL removes the default clear search icon that appears in
Tab Search's input field to align with other search field examples
in top chrome (Omnibox) and the NTP's WebUI Realbox.

Before:
https://drive.google.com/file/d/1Ts4JBBdPqD1IvNvD181VQVYFFm9bH4Kd/view?usp=sharing

After:
https://drive.google.com/file/d/1EYvqbL6OFB-672unHwZCIa21s6adAhvo/view?usp=sharing

Bug: 1147817
Change-Id: Ie8333ce6797e0ece91827af7cb1eaba521175d91
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2536139Reviewed-by: Yuheng Huang <yuhengh@chromium.org>
Commit-Queue: Thomas Lukaszewicz <tluk@chromium.org>
Cr-Commit-Position: refs/heads/master@{#827539}

chrome/browser/resources/tab_search/tab_search_search_field.html

@@ -51,6 +51,10 @@
     text-overflow: ellipsis;
   }
 
+  #searchInput::-webkit-search-cancel-button {
+    display: none;
+  }
+
   @media (prefers-color-scheme: dark) {
     #searchIcon {
       color: var(--google-blue-refresh-300);