I think that much of what you propose is already enacted in practice, at least for many federal employees who are not political appointees, etc. Elected representatives are another matter entirely.
Many federal employees do in fact have to limit what securities they can purchase or hold. For example, if I recall correctly, patent examiners cannot hold more than $20,000 in investments for an industry that they examine patents in, and they cannot hold more than $10,000 in stock for any company they may examine patents for. The guidance tells them to prefer index funds that capture the entire market rather than individual companies or sectors. This is explicitly to prevent conflicts of interest. If you are high up enough in these federal positions (without having to be a political appointee), you even have to complete a financial disclosure form (SF-714), and people do check these things for potential conflicts.
That said, many federal employees invest heavily in TSP funds, most of which are managed by BlackRock, so your last point about distributing them among multiple fund managers is not true in practice, since I imagine most federal employees have their investments heavily in funds managed by BlackRock.