Much of my career before I started my company was either as in-house HR or as a consultant/coach. What you are describing is very common and often is outsourced at your size/stage. Internal HR is usually buried in day to day operations to manage the architecting and building of communication, strategy and process. Also, those HR leaders senior enough to drive this and actually add value, tend to be very expensive and often too much for a small company to absorb on a full-time basis. A consultant can come in and then be available more adhoc until you can bring someone on full-time and then you are not starting from scratch.
As far CEO's having HR meetings, the HR leader/consultant also acts like an Exec coach, so yes they do, but perhaps not in the same way most people are used to.