Your problem is completely BA related. You're flying on a BA ticket, with BA rules and BA fare conditions. It seems like the PNR that you're using was created by BA and it doesn't matter what airline you're flying with you probably won't get 100% functionality. In majority of cases you will have enough control to add excess baggage, allocate seats and modify any APIS information that is required. However if you want any advanced control of your ticket you'll need the operator created PNR which can be up to two additional PNRs. Also what you would need to keep in mind is when flying on a codeshare you are not always entitled to benefits offered by the operating carrier as you are considered to be flying on the codeshare partner and depending on the codeshare agreement determines the benefits you get. If this is the case, which is what QF has told you it is, then it seems like it's a BA related problem rather than a QF MMB problem as QF can't limit something that's not there in the first place.
That's the issue you have when you have restricted dietary requirements. You either need to be flexible with what you eat or choose an airline that will give you something that you will eat. The airline has given you something edible, but if it's not delicious enough for your liking then choose a different airline that will give you something. I understand that you chose QF because you expected to be able to use Qeat, but you're not flying with QF, you're flying with BA. As mentioned above and by others you're on a BA ticket. If you want QF Qeat fly QF on a QF number to be considered QF pax.
Now there is possibly a way around this. You complain that QF hasn't given you access to Qeat, but you haven't clarified whether you're using the BA issued PNR or the operating PNR issued by QF. These are two different PNRs and there is a chance that once you use the QF generated PNR you might be able to access Qeat. The BA generated PNR will naturally always be restricted (even if it's created on the same GDS), whether you see that restriction or not depends on what you normally modify your booking with. Qeat is a little complex and convoluted and hence you might run into the perceived MMB restriction. If even after using the QF generated PNR and that doesn't work then your beef is with BA for not negotiating under the codeshare terms access to Qeat.
As for alliances, yes they're there to streamline transfers and connection but they are not there to make it feel like you're flying on one airline. Inter-airline differences will show and what alliances do is make that appear as minimal as possible. Alliances ensure that your baggage will follow you through because of mandatory interline agreements. Alliances will reward you to standardised extra baggage allowances and status benefits when flying on an alliance airline while using your home airline's status. But most of all, alliances are there to improve business for the airline by funneling pax between each other.