Broadcom is publicly listed with a public float of about 98% (i.e. 98% of it's shares are listed publicly).
You're right that most shares are held by institutions (~80%), but that typically reflects the fact that most share ownership by individuals/companies goes through intermediaries (401k, fund investments, ETF etc.). Most of this institutional ownership is just asset managers, insurance, banks etc. taking their cut before passing returns/loses through to the end risk holder. The average institutional ownership of companies in the S&P500 for example is also ~80%.
None of this takes away from the point that Broadcom is absolutely run like a PE firm as the original commenter noted.
Not surprising given the CEO was appointed by KKR/Silverlake 20 years ago.