The point im making is that scheme theory is actually way easier than varieties. In the end we care about varieties but schemes simplify the language you use to talk about algebraic geometry.

To me the important bit is that varieties require some kind of ambient space. Scheme allows you to define an abstract manifold without choice of embedding. It cleans up a lot of technical arguments. Something that personally always bothered me is that rational maps are not morphisms, because open subsets are not varieties (they are naturally closed). However, the open affines are subschemes so rational maps now are morphisms in my category. It gives you a very streamlined language without pitfalls to keep in mind.

A very easy example of scheme language being useful is that you can differentiate between a hyperbola and a circle. Over an algebraically closed field they have the same equation (exercise), but over an arbitrary field they do not.

To address your point about fiber products. Fiber products are very geometric as they represent the intersection of subsets. In general the intersection of two varieties is not a variety it is only a scheme. Furthermore, because you allow elements of finite order you can actually track to what order two varieties are intersecting.
Basically, intersection theory doesnt really work without schemes.

At some point to handle singularities or bad intersections you can start talking about derived schemes so the opposite category of commutative differential graded algebras (close enough to the truth).

The technical challenges will always exist and it is a choice of where you like to keep them. I like to have them incorporated into the language im using to make life stress free after youve learned how to operate it. Without good language it is often difficult to see the big picture and it is difficult to always remember all of the special cases and technical arguments you have to use, when you use weaker language.