From my long experience in helping Contact Form 7 users on the support forum I can say that the most common reason behind troubles is interference from other plugins, or with the theme which the user uses.
Some see this as an incompatibility issue between plugins, but that is not true in the precise sense. Plugin authors and theme authors don’t have to care about compatibility with a particular plugin — even if the plugin is one of the most popular ones. Authors only have to care about compliance with the best practice in WordPress plugin and theme development.
However, it is a fact that there are many plugin and theme authors who don’t care about anything. Their plugins and themes often interfere with other plugins, and can break their functionality. Many of them are commercial products, or products distributed under no proper licensing (CodeCanyon and ThemeForest are notorious as typical distributors of such poorly made products). No wonder you face problems if you use such plugins or themes.
This is the reason why you are advised to deactivate all plugins (excluding Contact Form 7), and switch your theme to the default theme at the same time. By doing this you can determine whether the trouble you face is caused by Contact Form 7 itself, is caused by one of the other plugins or is caused by the theme that you’ve just deactivated. If it turns out that the culprit is one of the latter two then by re-activating them one by one, doing the same test each time, you should be able to narrow down and eventually pin down the real culprit behind the trouble.
When we say “default theme” this refers to the official theme that has come with WordPress itself. If you use WordPress 5.3 then Twenty Twenty is the default theme. Twenty Nineteen and Twenty Seventeen are former default themes, and they are still maintained. Default themes are more reliable than other themes because they are created and maintained by expert developers.
You may worry that you might lose configuration data when switching themes. If the theme is created in the right way, and is compliant with recommended practices, then you can safely switch themes without any data loss. If not, unfortunately, there is no guarantee. In any case you are advised to make a backup of data beforehand.
There may be cases where you feel it is difficult to deactivate plugins, or to switch themes, because of reasons such as it being the client’s site and they might be unhappy if you were to do that. I appreciate the difficulty of these situations, but if the responsibility to troubleshoot lies with you then you have to do that somehow.
Many experienced webmasters, in anticipation of such cases, prepare a staging environment on which you can test changes without effecting the live production environment. Or you may want to try a plugin called Health Check & Troubleshooting. This plugin allows you to make changes to a site in such a way that the changes are only visible to you as the logged-in administrator.