Your arms are NOT designed for lifting or pushing. Might seem a bit odd to hear, but its the simple truth. Your arms are designed to re-direct energy from elsewhere in your body.
Take for example the pullup. or the chinup, or whichever orintation your more comfortable. Yes, of course the strength of your bicep does play a part. the whole system needs to work together. but primarily your bicep is acting as stability and a an aid to the levers that make up your arms so that your upper back can do most of the work.
When performing a pushup, the power typicaly comes from not just your tri's, but from your pecs, and shoulders. Spend a week or two doing nothing but pushups that concentrate on your shoulders and you'll see an amazing differance in the standard pushup.
In a self defence situation, throwing a punch. The power doesnt come from your "BiG gUnZZZZ". It comes from your koshi. Your core. The power of your punch is generated from the rotating force you can create by using the lower 2/3 of your body. All your arm needs to do is direct that power.
Even something as simple as picking up an object. The bulk of the work is done through your legs, and an assortment of levers inside of your body giving you mechanical advantage over the object your lifting.
If you want to be truely strong, dont waste your time on muscles that get enough work doing things that matter. If you weight train, stick to the big 3. If you do bodyweight, pullups, pushups, one legged squats and burpes. everything else is tweeking for optimal performance.