Why do publishers allow distributing preprints?

Although this is speculation since I don't work for a publisher, there are a couple of reasons I can think of:

• Due to pressure from funders, who want the research they funded to be publicly available.
• To take away the focus from gold open access. Publishing preprints is known as green open access, and means that authors are allowed to post their work elsewhere. Often, however, this is with restrictions: only the non-peer reviewed version is allowed, and only after a certain period. This discourages academic libraries from cancelling their subscriptions, and appeases the funders who might otherwise call for the published version to be publicly available immediately. (Note: I think this was mainly a motivation before the publishers figured out they could heavily charge funders for the latter requirements.)
• Because it might be hard to argue in a legal case for publishers to restrict access to research that they haven't been involved with at all (i.e. not yet been peer reviewed, layout done by the researcher themself, etc.). Yes, they often coerce authors in transferring copyrights to the publishers, but they might rather not test the strength of that argument in court.
• As a courtesy, to maintain goodwill. It's clear now that it doesn't really threaten their business model (no subscriptions appear to have been cancelled in response to preprints being available), so they can easily do this.

If a publisher doesn't allow distributors of preprints, they'll never get a submission from me. Why would I needlessly restrict the distribution of my own work? The publisher doesn't pay me anything in exchange for these rights.

Because it will happen anyway, so they might as well appear magnanimous by permitting it.

Authors want their work to be read. The publishers effectively have three options:

• ignore it, which makes them look as though they're not in control
• sue their authors, which would probably be futile and would certainly reduce submissions to their journals
• say that preprints are allowed, and focus their efforts on trying to convince people that it's worth paying for the "version of record" anyway.