Should a Search Icon Inside a Search Bar Be Tabable?

I'm developing a search bar. The input field and the icon are uniform in design, and thus a tabable icon feels superfluous because the default behaviour when pressing the enter key while the input field has focus is to submit the form. Hit tab>enter and you're achieving the exact same. However the icon is a clickable button, which I believe is a necessity. Should there be a 1:1 relationship between tabbing and clicking? Meaning if something is clickable, then it needs to be tabable?

Another challenge is the "X" left of the search icon, having the role of clearing the input. Because it has a behaviour different to the default one of the input field, I feel like it should be tabable. But at the moment it feels like I'm tabbing backwards when giving it focus. It also looks a little weird in its focused state.

Any input on this situation? Thanks.