(see the short version)
In case this is of use to other travellers...
I booked an organised holiday through a tour company (Exodus). I had the option of booking my own BA flight to Naples, or letting Exodus include them in my holiday booking. A quick check showed me that the prices would be pretty much the same. Considerations of potential liability/insurance claims led me to book the holiday inclusive of flights.
There followed extended exchanges of emails with both my tour company and BA as I attempted to confirm check-in arrangements and whether I could earn BA miles and tier points on the flight with my BA Executive Club membership. Exodus notified BA of my membership, and the flights appeared in my BA mobile app under 'my bookings'. Not, however, on their website when I logged in.
A helpful customer service assistant at BA checked my flights and tier points (20, the standard Euro Traveller amount), and a couple of days later the flights appeared on the BA website under 'My Bookings'. However, clicking on 'Manage my booking' led me to an error message.
Exodus assured me "You are able to check in on line 24 hours before departure" and erroneously stated that "You are unable to upgrade on this flight as they are small planes with no business or first class seats available" (I've flown this route and there is the usual BA short-haul Club Europe cabin). A week before travel, BA's flight search was offering two Club Europe seats on the flight at high prices... so probably unlikely any affordable upgrade will be on offer later. Regarding online check-in, BA wrote to me "As you have made a group booking, you will be unable to access your booking through 'Manage My Booking' and check in online for the flight".
Exodus suggest that specific seats may be requested through one of their agents - I may try this and report back.
Unless events prove otherwise, I am stuck with queueing to check in at the airport - with the August crowds - and getting a limited choice of seating. I was interested in upgrading the flights to earn more tier points, but this possibility is not open to me until the airport - minus any special offers that would have been accessible through 'Manage my booking'. An email from BA states ambiguously that "If you pay for your upgrade, you may then earn Tier Points for your upgraded cabin."
So booking through a tour company has given much less control over my flights, without the possibility to use online or mobile check-in, earn more tier points, or choose a seat prior to airport check-in.
As a note to anyone with the same initial dilemma: Yes, allowing your tour company to book your flights may well provide increased security if you have to cancel or if things go wrong. A group booking may be cheaper in some cases, and may be nice if you are travelling with a group you already know (perhaps groups are seated together?). But you may be sent back to the dark ages of air travel, without the convenience of queue-skipping online services. Frequent flyers will find this frustrating. And the upgrade/tier points restrictions may also be worth considering.
In short, if you want to keep control of your travel, think hard about how you book. I could have paid a similar price, travelled on the same flight, met the group as arranged in the destination airport, but benefited significantly if I had booked my own flights. It also, as it turns out, would have saved me the time spent trying to hunt down correct information about British Airways and group bookings. The conflicting messages and lack of online information has led me to write this article.
Perhaps there will be no airport queues (not even at Naples), a good choice of seats and no problems... I'll report back.