VCs are typically LLCs or LLPs (pass-through entities) with investors including pension plans, endowments, and other not-for-profit entities. Not-for-profit entities can make money from business or passive investments without paying tax, but must pay tax on unrelated investments. The sale of stock in a C-Corp is considered a passive investment and thus not taxable.
By contrast, though joining a partnership is not an income-generating event for tax purposes, it no longer becomes a passive investment and any income from the sale of the business would be taxable. Furthermore, this form of ownership would likely lead to phatom income (i.e., income that exists in the eyes of the IRS but not in your pocket). Let's assume a VC fund owns 50% of a company - if that company makes $100 but retains it for the operation of the business, it is still considered taxable income in a pass-through entity. As a result, instead of having an additional $30, the fund will actually owe $20.
A second major issue is that the owners of a pass-through entity cannot file their taxes until the entity files its K-1. If a VC invests in 50 partnerships, the partners in the fund couldn't file their taxes until each of those 50 companies first filed a K-1 - and it's generally not a good thing to have your taxes hostage to an entity which you do not have operational control over.