Is this really correct? I find it hard to believe that BBC Studios couldn't benefit from any off-screen sponsorship now that it's a commercial entity.
BBC Studios (the BBC's semi-independent production arm) can make sponsored programmes - sure.
But they can't make them for the main UK BBC channels. The rules about sponsorship vary by broadcaster, and so BBC Studios commissions from the BBC will be made under different rules to, say, BBC Studios commissions from Channel 5, or an overseas broadcaster, like PBS in the US.
The BBC does have sponsored broadcasts on BBC World News, but that's a non-public service channel and not covered by the same Charter/Ofcom rules as BBC One, for instance.
Rhetorical question, because I don't expect it to happen anytime soon.
What happens in the case of something like a international sporting event? The NFL have stated they want a London based team and that stadium would be treated like any other (as in, they can bid for the Super Bowl if they want to)
But lets just assume a London team exists, and the stadium they use is hosting Super Bowl 60.
BBC Studios would then be commissioned by the NFL to produce the coverage of the event for international use. The BBC would obviously commission them for the domestic coverage, and as Sky also have the rights, it would be possible for Sky to do the same.
Very hypothetical as the BBC don't have a sport division in BBC Studios - BBC Sport remained part of the public service element of the BBC for both production and commissioning.
So now you have three main problems to deal with. The BBC can't have any advertising or sponsorship. Sky can take advertising breaks (but not as many as the US can) and the US broadcaster can have a shed load of advertising and on-screen sponsorship (including things like the Ford pre-kick show, or Pepsi halftime show).
Pretty much the same issue that NFL have anywhere. The (non-existent) BBC Studios sport operation acting as host broadcaster would create a clean world feed without sponsorship elements, and possibly a dirty feed with sponsorship elements (it's normal to do a dual-mix with different replay wipes, stings etc. with different branding in these cases and modern vision mixers allow for this).
This clean world feed would be made available to third party rights holders, like the US broadcasters and the BBC domestic operation.
There may be a secondary feed produced by the US rights holders in addition.
As this coverage of an independent sporting event, not a BBC produced 'show' that is being staged by the BBC (i.e. the BBC didn't book the football teams to come and play) then this falls into an event category, where I believe the sponsorship rules are different.
The BBC domestic operation would have to be careful with the handling of sponsored pre-show and half-time content - and would no doubt take a world feed with less 'sponsored' graphic branding, and possibly also not take the US FoH MC sound if that were covered in sponsorship mentions.
However BBC Studios would have no issues - they are being paid to produce coverage, or an event, and they will do that to the spec provided by their commissioners (NFL in this case I expect) In that case they are no different from Done and Dusted doing the coverage.
Moving away from the hypothetical scenario you propose - if the BBC entirely produces a show, but accepts sponsorship this is an issue (c.f. the Robinsons sponsorship of Sports Personality of the Year being axed - as it was clear SPOTY wouldn't exist without the BBC) However, if the BBC covers a pre-existing sporting event, or takes a host feed from a sporting event, that has event sponsorship or advertising around the ground (like a football match) then the rules are different, as the BBC is not directly benefiting from that sponsorship.
Where the BBC are producing editorial coverage as party of a sponsored event like BAFTA, then the rules are different again, and the BBC is careful to restrict showing commerciallogos and producers are careful to script verbal mentions to an agreed limit.
Where Eurovision is complex isn't the on-screen branding - it's the payment for the actual contest. If large elements of that are sponsored in return for an on-screen credit or promotional consideration in surrounding content, that's going to always be tricky. There may be ways round it by the BBC surrendering some duties and responsibilities to the EBU - but it would be complex.