Since a single J6 liner is enough to convey information at its top speed, I will ignore the slower merchants. I don't dispute that backwaters will get their news late. What I'm concerned about is chiefly sector and subsector capitals.
It wouldn't be randomly determined. It would depend on available passengers willing to pay for high-performance service or belonging to organizations that they can persuade to pay for it. In other words, millionaires, Imperial officials, and corporate bigwigs.
For estimating total passenger numbers I'm using GT:FT. For estimating how many of them are bigwigs, I'm going with 1% of the total, which I'm quite sure is lowballing the figure tremendously. A case could be made for almost all long-distance (more than one or two jumps) travel being bigwigs.
You're overlooking the passengers that want to visit that world two or three or umpteen jumps away and don't care a whit for the intermediate stopovers.
Let's take a concrete example: Deneb to Mora. Deneb has a WTN of 6 and Mora one of 6.5. Distance is 20 parsecs. BTN is thus 6+6.5-2.5 (for distance) or 10. Passenger BTN is thus 11 due to Mora's status as a sector capital (I don't count it twice as both subsector and sector capital).
A passenger BTN of 11 means between 500,000 and 1,000,000 passengers per year or between 1369 and 2739 per day. This number should almost certainly be doubled, as, unlike goods, many passengers will eventually want to return home, so it's probably 1369-2739 Denebians and returning Morans going to Mora and 1369-2739 Morans and returning Denebians going to Deneb. But let's ignore that. If just 1% of the passengers (13-27/day) are bigwigs, you have a basis for a small high-performance Deneb-Mora passenger service that crosses 20 parsecs in 4 jumps without caring about intervening worlds.
(As it turns out, Deneb to Mora is a bit wrong as an example, because the optimum route happens to go via Vincennes, so you'd have additional passengers that only want to go from Deneb to Vincennes (and back) and from Vincennes to Mora. And on top of that you get every passenger from everywhere else in the Imperium who wants to visit Mora. I can't estimate just how many that will be, but the passenger BTN between Capital and Mora is also 11, so just from Capital you get another 500,000 to 1,000,000 per year. And I'm very much inclined to believe that more than 1% of them will be bigwigs.)
Anyway, I see no reason to believe there won't be commercial support for connections between high-population worlds that are quite distant from each other.
Subsidies may come into play in some cases. The Usani-Mora run would possibly not have enough passengers to support a high-performance passenger service, since Usani has a population of only 2 million[*]. A subsidized J6 connection to Hdr would allow bureaucrats from Usani to link up with the 17-parsec-in-3-jumps route to Mora (And the J5 to Vincennes route too).
[*] Though it's pretty likely that a much higher than average proportion of those 2 million are Imperial bureaucrats and corporate lobbyists and such-like. (or perhaps there are millions of lobbyists that don't count because they're transients and travel back and forth every few months 
)
)You may or may not have a point there. The Deneb-Mora example averages 5 parsecs per jump, which is 83% efficiency before taking handoff time into account. On the other hand, Usani to Mora is 23 parsecs in four jumps, an average of 5.75%. Vincennes-Mora is 6 parsecs/jump.
It would really be necessary to calculate each main route in the Imperium specifically. Back when I had a computer that could run Galactic, I mapped out a J6 route from Capital to Mora that had the occasional short jump to accomodate important worlds. Unfortunately I can't find the figures, so I can't say anything more about the efficiency.
Once a passenger liner arrives in a system, priority news can be transmitted to the mainworld in whatever time the available bandwidth will allow. After that, the information can leave any time an outgoing ship is sheduled to leave, quite possible long before the incoming ship reaches port.
I think handover times for the main news items would be much lower than your estimates. Bulk mail would be quite a different matter.
It would double the cost unless it was a sunk cost, the owning organization having to pay for those couriers anyway. In wartime, J6 couriers (not just J6 -- I'm sure the IN will have some couriers with lesser jump capacity) are needed in great numbers to collect and disseminate information. In peacetime, those couriers would be used for optimizing transmission of reports and orders for no other reason than that they're there.
With enough couriers, the IN could even use the multiple courier scheme to increase information speed. Dispatching two couriers at the same time to the same destination decreases the average transmission time by about 6 hour. Four couriers get the average down by about another 2 hours. Beyond that point the law of diminishing returns apply with a vengeance.
There's even evidence that the IN does employ such measures. There are three TNS newsbriefs datelined Regina in 1112 that quote dispatches that have been brought from Terra to Regina in 336, 375 and 375 days. I've estimated the number of jumps from Terra to Regina via Capital and Mora to be 56, for an average transmission time of 6 days 0 hours for the first and 6 days 17 hours for the two others. By assuming deep space fuel depots in strategic places I can shave 5 jumps off that, for averages of 6 days 14 hours and 7 days 8 hours. Alternatively, by assuming a direct run from Deneb to Regina (instead of detouring through Mora), I can shave off two jumps, for averages of 6 days 5 hours and 6 days 23 hours. But that includes handoff time.
My preferred explanation is no deep space fuel depots (or perhaps one or two), direct Deneb to Regina routing (54 jumps), efficient handoffs, double couriers, one extremely lucky run (could be a record) and two average ones. YMMV, but whichever options you choose, the IN quite evidently is very much on the ball when it comes to moving information.
Incidentally, those three newsbriefs are also proof that the IN carries news for civilians. I guess they don't prove that the IN also carry government dispatches, but I think that's a pretty good assumption.
Hans
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I hadn't looked at it awhile myself, thanks for bringing it back to my attention. The later Gal24 using visual basic, I think, would be harder to run. 
