Nashville, Chattanooga etc. were major rail junctions. If I may post the UMSA map on wikipedia:
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McClellan's grand strategy was to break up the rebel communications. His ordered operation against Ft Henry was to allow the destruction of the bridge south of it, disconnecting Beauregard's Army Corps at Bowling Green and Polk's Army Corps at Columbus. The destruction of the bridge at Clarksville would then disconnect that entire section from the Confederacy.
The question that then vexed Halleck was where would Beauregard go? McClellan and Buell believed, correctly, that he would fall back on Nashville. Halleck believed, wrongly, that he would move to Clarksville and fall on Grant.
McClellan, for his part, is looking at grand strategy and is pushing for, as soon as the bridge at Clarksville is gone, an expedition down the Tennessee to Decateur. This would have isolated the two armies, and prevented Polk and Beauregard forming a junction.
In fact, on 3rd January 1862 McClellan ordered the expedition up the Cumberland for the specific purpose of breaking communications between Polk and Beauregard, although McClellan expected it to be Polk reinforcing Beauregard.
Donelson didn't have 12,000 troops until this campaign. The force at Dover/Donelson was about 2,000 men, mostly because this was a recruiting camp (3 regts in the process of being raised there, 49th, 50th and 53rd TN). The actual fort had a regiment (30th TN) and Forrest's cavalry regiment was also based there. However, the defenders of Henry (about 2,700 in 5-ish regts* and a bty) fell back on them.
Then, Pillow has a visit from the good idea fairy. He decides he wants to attack Grant and starts sending the force at Clarksville, and additional forces being sent there (because this is what AS Johnston considers the enemy objective to be) to Dover. Clark's brigade is first, although Clark disputes Pillows orders and resigns instead of following them. Buckner with two brigades and Floyd had been ordered to Clarksville to bolster the defences there, and Pillow sends them to Donelson instead. Taking command on the 9th, his orders show 5 regts (3 from Clark - 1st MS, 23rd MS and 7th TX, and 2 from Floyd - 51st VA and 56th VA) had been forwarded to Donelson. Six regts from Buckner (2nd KY, 3rd TN, 8nd TN 26th TN, 32nd TN and 26th MS) arrived the 9th-10th. Reaching Clarksville, Buckner puts a stop to this. Together with Floyd they stop the reinforcement of Donelson and instead start constructing a new position at Cumberland City.* On the 11th Buckner arrives at Dover to begin the removal of the reinforcements diverted by Pillow. Pillow objects and goes to see Floyd. Floyd agrees with Buckner and has been trying to get AS Johnston to approve not making a determined defence at Donelson. However, Pillow is angered by this, goes to see Floyd and basically browbeats him into putting everything into Donelson to attack Grant. They really don't start properly digging in until the 11th-12th.
Strength, 7th: ca. 5,000
Reinforcements arriving on the 8th: ca. 1,900 (hence the estimate of 7,000 late on the 8th)
Reinforcements arriving on the 9th and 10th: ca. 3,800 (about 10,700)
Reinforcements arriving on the 13th: ca. 3,000
My point being that when Grant was considering pursuing to Dover with just infantry and cavalry, there wasn't a huge force there.
As of 8th February, when Grant had already decided Donelson was too strong, there were only about 6-7,000 men there. However, Halleck was worried about the whole of the Bowling Green force striking Grant, who he now considered to be exposed, and so arranged to move on Clarksville via Dover. He wasn't really expecting Pillow to have diverted the Clarksville defenders to Dover.
Anyway, I started writing that weeks ago, and got hung up on when arty units was at Donelson which, as you can see, isn't resolved, but I'd best hit post being windows updates.
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Richmond was a major objective because it provided a huge percentage of rebel war material. Ultimately it was abandoned because Sheridan's force managed to cut the last railroad. Harper's Ferry was always an objective because it controls communications.
It all comes down to logistics.
I was making a
reductio ad absurdum.