When thinking about alumni and university governance I went back and read the relevant sections of Henry Hansmann's "The Ownership of Enterprise". Hansmann make a couple of interesting observations about university alumni. The first is that (partial) control by alumni makes sense if there is a ranking of universities based on perceived quality of students. Top tier universities would have market power and could charge a very high fee to gain entry into them. A profit maximising university would do this, but a university, at least in part, controlled by its alumni is less likely to do so as the alumni do not want to charge themselves a monopoly price. The alumni may have children or grandchildren going to the university and would want to keep cost of attending the university 'low' - or at least lower than it otherwise would be. Such an advantage of alumni control is less of an issue here is New Zealand given the amount of government control over universities and their funding.
The second point made by Hansmann relates to the use of "implicit loans" from the university to their students. The markets for loans for the acquisition of human capital are inadequate for obvious reasons. One way around this is for the universities to provide a crude substitute for these loans. Universities supply education to many students at below cost in return for an implicit commitment on the part of the students that they "repay" their loan through donations during their life after university. One result of the government supplying a system of loans, grants or loan guarantees is that there is less, or no, implicit loans to be repaid. This may one reason for the low level of alumni involvement in universities in New Zealand.
The cost to the university of alumni involvement for either of the reasons above is that the alumni will want a share in control of the university. Thus if the universities wish to gain financially even more from their alumni, as they seem to want to do, then they will have to give a larger amount of control over to the alumni.
As noted in the previous post on this subject the alumni can have the incentives and knowledge to make them a sensible class of patrons to give control over to. Not the universities or the government are likely to see it that way!