Should I use an Index when the "name" column never changes but the "views" column is frequently updated?

Question

I have a database table with 9254 rows in it.

These rows are all searched primarily using the "name" column on a website using php.

I was thinking of using an index on the name column to reduce load but I also have a "views" column that is updated whenever a name is searched on the front end.

So if someone does a SQL query of:

SELECT * FROM namestable WHERE name = "John Smith"

then an immediate second query is run:

UPDATE namestable SET views = views + 1 where name = "John Smith"

But also, on the result page for that query. This is then run as a seperate php file call (MySQL Full Text Search):

$sql = "SELECT name, id,
           MATCH (name,description) AGAINST ('$nameplus') AS score
      FROM namestable
      WHERE MATCH(name, description) AGAINST('$nameplus')
        AND name != '$name'
      group by name
      order by score desc LIMIT 8";

Would this directly impact indexing the name table as I read you aren't supposed to do it on rows that are frequently updated or that use MySQL Full Text Search.

I also have a page that calls almost every column in the database (to be used as front-end filters for results). So I'm honestly not sure if I should index each one of these columns as most of the content in them will remain static once set. The only one that updates constantly is the views column.

EDIT: Added more info.