Why are most universities reluctant to offer open-ended contracts to research-funded-staff (such as postdocs)?
Using the term "postdoc" may cause confusion, because to me a postdoc is by definition an early-career researcher hired for a limited period.  But the question as I understand it has nothing specifically to do with the terminology.  It amounts to why universities don't offer long-term contracts conditionally on maintaining adequate external funding, but rather offer successive short-term contracts.

One answer is that sometimes they do offer long-term contracts.  In U.S. academic medical research, it's common to have long-term "soft money" positions for which most or all of the funding comes from grants.  There are even tenured soft money positions, in which you have a guarantee that you cannot be fired without good reason, but where there is no obligation for the university to pay the salary except as supplied by external grants.  (Sometimes they are responsible for a small percentage of it, and sometimes none at all.)  This becomes less common as you move further from medicine, but long-term soft money positions exist across science and engineering in the U.S., at least in small numbers.

Why not always do this?  One reason is to avoid raising expectations.  My impression is that it's rare to offer a long-term soft money contract unless there's a reasonably high chance that money will continue to be available for quite some time.  Even if there are no legal obligations, it's terrifically awkward to fire someone from a position for which they mistakenly believed funding would continue, and it's easy for people on soft money to be irrationally optimistic.  (You see this frequently with tenure-track jobs: someone starts by recognizing that they genuinely might not get tenure, but after a year or two their nerves have settled and they find it tougher to take the possibility seriously.)  This means there's a strong incentive not to offer long-term contracts that are too speculative.

Another issue is whether the department or university really wants this person to have a long-term affiliation.  There's sometimes a feeling that people on short-term contracts don't really count for the department's reputation, since they are just visiting anyway, while people on long-term contracts count more.  This means other faculty members have more of a reason to exercise oversight.