The short answer is no, early humans and other extinct hominoid species did not spend most of their lives in caves or rockshelters. Most archaeology sites are found in open-air settings (near important resources). These sites often lack good organic preservation and do not get the same amount of attention as cave sites that can contain very diverse and unique assemblages. The heavy focus on these extraordinary cave collections is likely responsible for creating the misconception that early people primarily lived in caves.

My dissertation work focused on the settlement-subsistence strategies of Paleoindian/Paleolithic groups in the Intermountain West region of the U.S. One of the issues we face in this region, is that much of what we know about Paleoindian subsistence and perishable technology comes from cave and rock shelter sites. While the dry setting of these sites create a perfect environment to preserve organic material, research suggests that these cave occupations were primarily short-term occupations that focused on a variety of specialized tasks (e.g., caching, resource processing, hunting camps) and that they only offer a small snapshot of what life was really like for these groups. There is little evidence to support the use of caves and rockshelters as long term residential camps in this region. Instead, we find the vast majority of pre-contact sites in open-air settings. However, because open-air sites in desert regions are often exposed at the surface, most of these sites consist solely of stone tools. Cave and rockshelter sites on the other hand can contain wood, bone, textile, and other perishable artifacts that are very “sexy” to both archaeologists and the general public because of their research potential and rarity. Given that these sites are so rare, they often generate the most academic attention and, in turn, are the most likely candidates to be shared with the general public. This can lead to a misperception by the general public that early people primarily lived in caves. While this is not the only reason for this misconception, it may be one of the primary drivers.

Ultimately, caves and rockshelters are great places to get out of the elements and to store food/gear (among other things) but most humans like to live close to food, water, and other essential resources that are often found far from caves.

You might consider /r/Askanthropology for this question.

Additionally, while there's always more that can be said, this comment by /u/pachacamac may be of interest.

Loved reading all the responses. Thank you for this.