My understanding is that only judges can share these recordings because it would be an ethical violation for the lawyers to do it, but I wonder, why isn't it an ethical issue for the Judge?

It varies from court to court, but in general the judge has the responsibility of evaluating whether public interest outweighs rights to privacy for any given piece of recorded court information. I'd imagine it is an ethical issue, but the judge has the authority to evaluate the ethics and reach a decision.

I would imagine it is harder for the judge to benefit from sharing recordings, where a lawyer could use a release recording to promote their practice or attempt to sway public option in the event of a mistrial.

Indeed. The adversarial court system means that every lawyer involved is on one side or the other; the judge is both neutral and in charge overall, so it makes sense for the judge to make that decision.

Well in ordinary circumstances it is often allowed for the public to be in the courtroom (not always, obviously), so I would think that the judge would apply the same criteria? IANAL.

Nominally (before Zoom court) court proceedings (and records) are open to the public.

Why are the cases being streamed on the internet?

Criminal court proceedings are normally open to public by default in many jurisdictions.

The idea being that law is applied in public to keep process in check, unless there are exceptional reasons not to do so.

...and with public viewing galleries in courts closed or access-limited due to covid, live-streaming over the Internet is the straightforward replacement.

Though there is a substantial difference between having an modestly-sized, in-person audience of dispassionate local journalists, plaintiffs, relatives and the occasional curious member of the public - and having potentially thousands of Internet gawkers and potential doxxers. Especially in cases like rape/sexual-assault and the like: there's definitely a public-interest (for the accused, the victim, the guilty and the innocent alike) in having a limited audience.

...can/would any judges agree to limit the public streaming audience to a smaller number in those cases?