Which everyday components involve flows of charge that are not electrons?
Now, this does get confusing when you get to semiconductor theory, and I understand your issue. I can name one very important case. When working with charge pumps in the human body. Many places in biology the charge flow is positive. When taking a biomedical modeling class for EE we often had positive charge flow.

We can get crazier, what if you have cancer? There are many options, sometimes you pick radiation. Photon radiation exists, what about proton radiation? The amount of protons they are sending are measured in Amperes. Why? Positively charged particles per second(enjoy the pun).

The important part here your particle makes an issue of. If electrons were positively charged the issue would be swept under the rug by most people. The fact that they are negatively charged makes people think about what it really means. 

If you really get down to the physics it is just a sign convention and is a menial problem. If you would like to assign them positive charge please do so, be internally consistent and do not publish anything and no one will be the wiser.

The most important thing is that if electrons were positively charged we would not have nearly as great of a name for the positron. I personally would not live in a world were the negatron is a particle.