Are there universities where authors can/must transfer their copyright to the university for negotiations with publishers? Say you are a researcher employed by a university and publishing your work in a closed-access journal, which asks you to sign a copyright transfer agreement. To simplify, let's say you are the only author. In most cases I am aware of, the researcher owns the copyright to the article, and so end up signing the copyright transfer agreement with the publisher themself.

However, in a sense this is curious: publishing your research is part of your job, your employer may want to have a say in what you sign, and they could also provide legal assistance to understand the implications of what you are signing. Indeed, publisher agreements often contain pretty wide-sweeping language (e.g., taking responsibility for errors in the work and subsequent damages), and it's weird that researchers would sign them, especially when they are not lawyers.

So I was thinking that it could be a useful service if a university had an office for this kind of legal matters, providing standard paperwork for their researchers to transfer their copyright to the university, which would then deal directly with the publisher. It could even make sense for the university to require its researchers to only perform copyright transfers via them and never directly, so that they can keep an eye on what the agreements are about, negotiate standard conditions for the publication of their research, and maybe use them as some kind of argument, e.g., when negotiating transformative publishing agreements.

Are there universities that provide such a service, or have such a policy?

The closest thing I can think of is how some researchers, e.g., federal employees in the US, have a different publication copyright process because their work is ineligible to copyright; but this is a somewhat different situation. Another related situation is the University of California requesting permission from their researchers to archive their works, but it does not look like they are involved in the copyright transfer agreement with publishers (except to notify their policy to them).
So I was thinking that it could be a useful service if a university had an office for this kind of legal matters, providing standard paperwork for their researchers to transfer their copyright to the university, which would then deal directly with the publisher.

I think the approach you describe would be rare. There are certainly institutions providing some phrasing to enforce their copyright/open access policies to the author, and some that have a copyright office that can take a look at a given copyright agreement and OK/veto it. I'm not aware of a place that would directly deal with the publisher. Indeed, given the typical journal system with a single corresponding author it is much simpler to give the author permission to sign it.

It could even make sense for the university to require its researchers to only perform copyright transfers via them and never directly, so that they can keep an eye on what the agreements are about, negotiate standard conditions for the publication of their research, and maybe use them as some kind of argument, e.g., when negotiating transformative publishing agreements.

These goals can be achieved without having the university deal directly with publishers, by instituting a policy and providing guidance.

Are there universities that provide such a service, or have such a policy?
The closest thing I can think of is how some researchers, e.g., federal employees in the US, have a different publication copyright process because their work is ineligible to copyright; but this is a somewhat different situation. Another related situation is the University of California requesting permission from their researchers to archive their works, but it does not look like they are involved in the copyright transfer agreement with publishers (except to notify their policy to them).

As I suggested above, it'd be rare for an institution to be directly involved and to require intermediate copyright transfer to them. However, there's an increasing number of institutions with open access policies, which require a non-exclusive license, and then copyright agreements with publishers to respect said license. (Waivers may be issued for some exceptions.) The policy of University of California appears to be of this type:

The UC open access (OA) policies reserve rights for UC faculty and other employees to make their articles freely available to the public in an open access repository. They do this by automatically granting a non-exclusive copyright license to the University prior to  any later agreements authors may make with publishers. UC retains those rights regardless of what rights authors may subsequently transfer to publishers.The OA policies don’t say where UC authors should publish or require them to pay open access fees to publishers in order to comply.

They direct questions about copyright or the policy to Office of Scholarly Communications, and provide an addendum that can be used to inform publishers about the policy.  Princeton similarly has a Copyright consultation service, and a similar open access policy:

On September 19, 2011, the Faculty approved an “open-access” policy that is intended to make the faculty’s scholarly articles, published in journals and conference proceedings, available to a wider public. Under this policy, faculty members grant to the Trustees of Princeton University a nonexclusive, irrevocable, worldwide license to exercise any and all copyrights in their scholarly articles published in any medium, whether now known or later invented, provided the articles are not sold by the University for a profit, and to authorize others to do the same. This grant applies to all scholarly articles that any person authors or co-authors while appointed as a member of the Faculty, except for any such articles authored or co-authored before the adoption of this policy or subject to a conflicting agreement formed before the adoption of this policy.
To avoid a conflicting transfer of copyright to the publisher and to protect yourself from breach of contract, you can attach an author addendum to your agreement with a publisher. The addendum is specifically designed to deal with the prior license granted to the University.

As a final example, National Laboratories under US Department of Energy have to follow a similar open access policy from DOE. Note that researchers at such institutions are often employees of a federal contractor, not of the federal government directly, so they can create copyrighted works. I know that at least Oak Ridge requires authors to put a &quot;Notice of Copyright&quot; on each manuscript sent to a journal to  inform them about the policy, and that they have a copyright office that will OK/veto signing a given copyright agreement with a journal.