I'll toss my hat into the fray....
You asked two important questions; a) What does having more gears in the transmission change and b) How does that affect axle gearing.
As most would agree, OEMs spend MILLIONS of dollars and thousands of hours on R&D prior to product launch, so everything said in this thread is really arm-chair, this included - but math is still math
With regard to transmission, when I was a pup, you had three speed autos and 4 speed manuals and gearing was somewhere between 3.08 and 4.10 in most every PU regardless of capacity. So what changed to cause more gears in the trans? Speed limits first and Fuel Economy!
3/4 gears covers the 0-55 MPH operating range pretty well, by the '80's 4 speed boxes were standard, which works out to ~14 MPH/gear. Of course even then FE was not a BIG concern for anyone despite the oil embargo of the mid '70's. Thru the '90's, CAFE was applied to pickups, specifically those at 8500# BVW and under, yup, that's the number. Over that GVW rating, the truck is considered a work vehicle and was not subject to CAFE reporting. I do not know if that number has changed, but if not, it would explain the severely reduced capacities on modern PUs that typically weigh about 6k-6.5k pounds in 4 door 4x4 configurations; it keeps then IN the CAFE reporting range. Now that speed limits are 70-80 MPH, 4 gears will not get the job done, because you still need 1st to get it moving and "OD" for highway cruising - that only leaves TWO gears to cover the 10-60 MPH spectrum
With 6/8 gears and a speed limit of say 75, you average ~11 MPH/gear. Importantly this does two VERY important things: 1) Keeps the engine speeds under control with smaller steps between gears, this leads to better performance and longer life and 2) Allows calibration of the system to optimize both pulling power AND fuel economy by electronically adjusting shift points to very closely match engine load. You would "short shift" at light loads keeping engine RPMs down and therefore volumetric efficiency high and "power shift" at high loads to keep the engine at peak power, which is also enabled by the smaller steps between gears.
As for axle ratios, you can only effectively add so many "OD" gears and without adding serious engineering and material cost, two OD gears is considered a practical maximum. If you need to keep the engine at 1800 RPM at cruise speed (say 75 MPH) and you have a top gear ratio in the box of 0.64:1 (TG - 1 = 0.822, TG - 2 = 1:1) and typical 17" tires with a Rolling radius of 31" which equates to about 640 Revs/mile.
As 75mph = 60mph*1.25, 640 Revs/mile (tire) turns into 800 RPM (tire) @ 75 MPH. Since RPM (engine) = 1800 and trans ratio = 0.64:1 (actually 1:0.64) we know Driveshaft speed is more than the engine speed (hence "OverDrive", or 1800/0.64 = 2813rpm(ds). What axle ratio will allow 2813rpm(ds) and 800rpm (tire)?
That is: 2813/800 = 3.52:1
Engineers choose the ratio for 1st gear and high gear based on mission requirements , then optimize how many are in between based on cost/benefit, RAM chose 8.
The peak Tq available to the axle shafts is a function of: Peak Tq(engine)*TC(typically 2.3:1)*ratio is low*RAR. We can see that assuming 440lb-ft peak (eng)*2.3*1st gear* = driveshaft Tq. DS Tq * RAR = Axle Tq > With a stated 3.21 axle Tq this is 5060*3.21 = 16243lb-ft, with a 3.73 this becomes 18874lb-ft!
The AXLES are only so strong, so engineers need to limit the torque to them, selecting rear axle ratios that will not exceed design limits of the axles is one effective method.
