Rule 53 states
Except as otherwise provided by a statute or these rules, the court must not permit the taking of photographs in the courtroom during judicial proceedings or the broadcasting of judicial proceedings from the courtroom.
This is applicable not just to the Supreme Court, but to all federal courts. Rule 53 applies to criminal cases, not civil suits.
Over time, Rule 53 has been expanded. It has changed from merely "photographs" to include television and related cameras.
The Judicial Conference of the United States prohibits the televising, recording, and broadcasting of district trial (civil and criminal) court
proceedings. Under conference policy, each court of appeals may permit television and other electronic media coverage of its proceedings. Only two of the 13 courts of appeals, the Second and Ninth Circuit Courts of Appeals, have chosen to do so.
So courts have the ability to choose whether or not they want to have their proceedings broadcaster/televised. It is technically allowed in the Supreme Court by all other legislation . . . but the decision ultimately rests in the hands of the justices.
In the 109th Congress, five bills appeared that would change existing rules, at some level:
- H.R. 1751
- H.R. 2422
- H.R. 4380
- S. 829
- S. 1768
H.R. 4380 and S. 1768 would apply to the Supreme Court; the others would apply to federal district and appellate courts. Both bills would make television coverage is mandatory, unless the justices collectively vote against it. So, once more, the power rests in the hands of the Court.
H.R. 4380 is brief, and would be merely an addendum to Chapter 45 of title 28, United States Code. S. 1768 is also brief, and would amend the same statute.
Title 28, in all its glory, can be found here (Chapter 45). The proposed additions are noticeably absent, as § 678 is not there.
So, addressing some quotes from your question,
Are there any reasons against cameras restricted to the Supreme Court, but that don't apply to courts that already allow cameras (such as the UK Supreme Court)?
There are no special laws, no.
To wit, does the allowance of cameras in other highest courts refute arguments against cameras in the Supreme Court of the United States?
It's purely the decision of the Court.
The rationale of the justices themselves against cameras is outside the scope of the law.