Basically yes, you’re right.

There are billions of bacteria all around you including inside you. Most bacteria smell because they simply produce some "smelly" compounds. Even yogurt has some smell, not to speak about E. coli.

For the same reason as they don’t smell e.g. lion or other predator which must actually catch his prey. Because one cannot smell anything against the wind. That’s a simple rule for all running around in the wilderness.

The human skin is quite a harsh environment for bacteria and other microbes: it is dry, low on nutrients, its oils and enzymes have antimicrobial properties and it is constantly being shed, which makes colonizing the skin difficult. Thus regular washing rarely gives room for harmful microbes, because the ones that have specialized to live on human skin have occupied the available living space in this quite unfavorable environment. So, for the "bad" bacteria to flourish, one needs to have some predisposing condition (like severe atopy etc.) or one has to wash themselves much more often than is normal. A very sensitive skin that is easily rashed or broken could be a risk factor as well.

The lack of water is perhaps the biggest obstacle for bacterial growth on the skin and thus tropical or otherwise moist and humid environments drastically increase the chances of skin infections or overgrowth of bacteria.

I think our ancestors were pretty stinky by our modern standards and like JackBean said, had to sneak upon their prey against the wind, pretty much as other predators do as well. Even modern humans are probably very smelly by animal standards and most animals can easily smell a human being whether they have bathed or not.

Normal sweat is pretty odorless, but in some regions (e.g. the arm pits) hormonal effects make us secrete many kinds of oils and pheromones and whatnot, which constitute much of the perceived smell especially when they interact with the skin bacteria. Many have probably noticed that when we are really nervous we not only sweat easily, but the sweat often has the typical, pungent smell on it that is absent in normal cases. Our ancestors probably relied much more on these "smell signals" back in the day, which the modern human is so desperate to remove by frequent showering.

Body odor is a perceived unpleasant smell our bodies can give off when bacteria that live on the skin break down sweat into acids – some say it is the smell of bacteria growing on the body, but it really is the result of bacteria breaking down protein into certain acids. Body odor can have a nice and specific smell to the individual, and can be used – especially by dogs and other animals – to identify people. Each person’s unique body odor can be influenced by diet, gender, health, and medication. Experts say that pheromones subconsciously influence how we react to each other; one human may find another sexually attractive because the smell of their pheromones is attractive to him/her.

Interesting information! Thanks!