It is always best to avoid the use of plugins as it is simply an overhead on server resources. I strongly recommend that you use header.php or functions.php for adding the code.
This is not true.
In the end, it all comes down to PHP code. If the code lives in a plugin, a theme, functions.php or you manually edit a core file doesn’t really matter for performance.
A problem can arise, when you install plugin X and that plugin brings with it a lot of functionality that you do not need. This can really slow down your site because there exists lots of code that is not needed.
But when you use (almost) all the functionality a plugin provides, why should your custom implementation be any faster? Additionally, I think you’re missing one very important part for this specific case: With proper (page) caching, for a visitor, the page is just as fast in any case.
One very important aspect you’re missing, as you’re only asking for performance, is maintainability.
Yoast has a team of developers, constantly working on their plugin. They can do much more than a single developer can, just because of numbers. Now say, Google changes something in their ruleset, for example the description must not be longer than X.
How long does it take a single developer to know this change happened and take the time to implement the changes?
When using Yoast, you don’t have to worry about that, someone else does. And you just need to click one button.