There are things an army can do which aren't attacking, and the Richmond Road is the route by which one could cut off the Richmond road.
Sure, but if Franklin thought Burnside wanted him to be doing something else, this was the first time he was hearing about it.
I don't think this is in accordance with the language Burnside uses
I would disagree.
Whole command is to be in position for a rapid movement down the old Richmond road. A division, at least, is ordered to go below Smithfield (i.e. downriver of Smithfield) and to seize, if possible, the heights near Captain Hamilton's. It's said that their line of retreat should be kept open - which shouldn't need to be specified if the whole corps is supposed to be marching just a mile away.
It's specified because that division is meant to be the spearhead of the attack for the rest of Franklins command, or as Frank O'reily says in his book on the campaign: "Burnside had ordered an attack on prospect hill to be supported by the rest of his command in force."
Franklin puts his whole command on the old Richmond road (that's 1st Corps, 6th Corps and 3rd Corps elements). He also makes a serious attempt to take the heights near Captain Hamilton's, which he does by means of an assault delegated to Reynolds.
I'd hardly call it a serious attempt. Meades division is really the only one that makes an progress, Gibbons and Birney hang back in support roles whereas Doubleday just watches the left flank. And yes, while Franklin basically had all of his men in line of battle he failed to real engage any of them, for instance Baldy Smith's whole corps took less causalities than Meades single division.
If what Burnside wants is for Franklin's whole command to pass below Smithfield, why only single out the "a division, at least" section?
It serves as a spearhead for the others, William Marvel speculated that Burnside most likely wanted Meade's division to neutralize prospect hill (which Meade felt his division would be sufficient to do) while the rest of Franklins command executed a turning movement after the fog lifted.
That's not what Franklin ordered, though? Franklin moves 6th Corps onto the Richmond road above Smithfield and 3rd Corps below it, while 1st Corps leads the attack with 3rd Corps able to support; Reynolds screws up committing anything beyond the first wave so the assault has nobody able to commit troops, but the only units who can really be described as guarding bridges by my reading of the positions is Burns and perhaps the right flank of 6th Corps. All of 1st Corps, all of 3rd Corps and the left flank of 6th Corps isn't.
Fair enough, however a movement that was supposed to involve the entirety of his command only winds up using a third of the troops at his disposal, and of the 20,000 men in the first corps, the only ones who really see intense fighting are Meade's and Gibbons. Now, if you want to blame Reynolds for failing to carry out the orders and not Franklins that's fine, but Reynolds at the very least understood their objective was to take the heights.
If I tell someone to be ready to do something tomorrow, that's not the same as telling them to do something tomorrow. The extra words involved in saying "be ready to XXX as soon as the fog lifts" is distinct from just saying "do XXX as soon as the fog lifts."
We know that warning orders are a thing - that is, you tell someone "be ready to attack the bridge" (just for example) and then you later send a triggering order to actually attack the bridge. This isn't an alien concept, and semaphore exists (the Army of the Potomac has a signal corps) so the idea of someone sending a short order amounting to "go now" wouldn't exactly be improbable - especially if the fog will have lifted by that time, meaning that Burnside has the ability to control the go/no go of Franklin's column pretty much in real time according to the situation on the right flank. It's about the only way he has to coordinate the two wings of his army, in fact!
That's all fine to say, but nowhere in that order does Burnside imply that Franklin should expect another order, he simply says to be ready to move when the fog lifts. Reynolds had telegraphic communications available for Burnsides headquarters, but when orders for the rest of the command to move never arrived, he never inquired if they were forthcoming. Later in the day he would actually refuse to act on Burnsides orders to assault the heights immediately to his front, as he said all of his men were engaged (Despite having two entire corps in reserve.)
Really, if what you're aiming for is a flank march around the enemy (that is, if we assume that your reading is correct), you almost want there to still be fog, to conceal the movement to at least some extent and delay the enemy reaction. It adds friction, it's true, but with most of the rest of the force not going below Smithfield (according to the orders) until the fog clears then it's materially delaying the flank movement relative to the individual movement (the attempt to take Hamilton) that's meant to take place still in fog. It means that Franklin's flanking wing has to march their head of column from above Smithfield (as the flanking wing is not the part that's going below Smithfield) along the Richmond road, then... to where? The order doesn't specify and there are several options, anything from attacking Hamilton from the east to attacking from the southeast to marching all the way to Grace Church before turning in to take the whole Confederate army in the flank and right rear.
A flank around the enemy is exactly what Burnside is intending, but upon going back to O'reily's book to look over this information, I realize I've been incorrect about something.
Burnside never intended for his men to march up the Richmond road you have marked on your map, he wanted them to attack here:
On Burnsides map, that causeway to captain Hamiltons I have marked in red was labeled as the old Richmond Road, it was apparently also below Springfield on that map. This is where Burnside intended Franklin to strike.