Neutral does and should carry normal currents
This may be the crux of your misunderstanding.
240V loads are carried on the two hots. 120V loads are carried on one hot and neutral. So a typical dryer might draw 20A of 240V and 3A of 120V. In that case the amperages of L1, N and L2 will be 23, 3 and 20. This is normal.
Ground is a safety shield only
The design philosophy of grounding is that grounds should never be used to handle current.
The circuit's overcurrent protection already protect from a fault between conductors. But what about faults between a conductor and anything else? That could cause a lot of mischief. Ground's job is to be the path of first resort for any wayward fault current. That works really well in metal-chassis machines.
"Wait. Won't GFCI do that same thing?" Kinda. Yeah. 30A 2-pole GFCI breakers can be had.
Neutral and ground should be separated everywhere except the designated Neutral-Ground Bond.
In an ideally wired panel, all neutrals land on the neutral bar, and all grounds land on the ground bar. There is a neutral-ground bond that can be removed.
This panel can be effortlessly converted to a subpanel by feeding it from another main and removing the neutral-ground bond.
It is also easy to measure ground fault current, by clamping a meter around the N-G bond.
However builders also have a lobby with the NFPA, and they lobbied for rules that let them spam it all onto one bar. It does no real harm, since they're connected anyway by the neutral-ground bond. It is leeeegal, just is a builder grade shortcut, and obviously prevents any of the things I mention above.
Why not bootleg ground on NEMA 14?
Well, that is what the NEMA 10 type hookup already does, so if that is your intent, there is no need to bootleg NEMA 14... you are already there.
It's back to the same problem as NEMA 10 and any other bootleg ground - any trouble with the neutral wire will result in the neutral being pulled up to 120V by loads, and with it the dryer chassis.
Whereas that does not happen if the neutral and ground are separated. In that case the machine just doesn't work and the ground continues to protect against hot-ground faults. Since there should be no neutral-ground faults, this becomes a non-event except the machine doesn't work obviously. The broken machine motivates the person to seek a proper repair.
If there was a neutral-ground fault, the machine may work normally misusing ground as current return. That is why bad repairmen create neutral-ground faults, to git-r-dun and onto the next job, safety be damned.
You can retrofit ground
Under NEC 2014 (and earlier for dryers) it is legal to retrofit grounds. You can run just a ground wire (#10) and retain use of the existing wires. It can go back to that service panel, or to any other ground also served out of that panel, as long it is of sufficient size (#10).
This same rule also makes a dryer ground retrofit useful for retrofitting grounds almost everywhere else, so it can be wise to plan it as a "backbone" for other ground retrofits.
My panel is full. How do I add circuits? is one of our most popular questions. The underlying reason is somebody tried to save a couple bucks when selecting a panel. In other words, what you're doing right now. As such, we Very Much Implore You to radically oversize the panel. It's cheap now.
Having a shutoff switch at the outbuilding is mandatory. Most people select a main-breaker panel and use the main breaker as the shutoff switch. Your way is fine too, and will save you a walk back to the house. I'm less thrilled with the equipment being outside though. I like equipment that's inside.
However, that switch is driving me crazy. From extensive Web search, the switch looks like a 3-pole switch, but the text all says it's a 2-pole switch. If it's got 3 poles, it's better to switch neutral. However neutral must always be switched with the hots, because if only neutral is severed, that creates even worse problems! That means a gang switch is fine, a 3-pole common-trip breaker is fine, but fuses are not. If using fuses, don't fuse neutral. Really there's no reason to fuse anything at this switch, the breaker in the house protects.
You do need a ground rod also. However you need two ground rods unless you do a really weird test to affirm one will suffice. Folk advice is just set another ground rod. Really, if lightning is around, you can't have enough ground rods.
Neutral is never ground
A lot of people struggle with the neutral-ground relationship and when/why they should be bonded. It's not random/whatever. It has very specific purposes and I'll hit two of them.
To keep all conductors near earth potential, i.e. so your hots are 120V from earth instead of 2400V from earth. It would be fine to use a 2-volt 5KVA transformer as your neutral-ground bond -- at which neutral would be intentionally biased 2 volts AC from ground. You could use a car battery for the N-G bond -- in which case neutral would be biased 12 volts DC from ground. We only use a strip of copper for a 0.0 volt bias, because it's cheaper and lower maintenance. There's nothing magical about 0.0 volts, in fact, neutral will not be 0.0 volts from ground anywhere but at the N-G bond.
To return ground fault current to source, which is the neutral line off the transformer. If something breaks and shorts 500A from hot to ground, that 500A won't flow (it'll just electrify things) unless we are able to carry it efficiently back to supply neutral -- in which case it will flow and will trip the breaker, as intended. Now imagine your main panel N-G bond fails, and you bonded neutral and ground at your outbuilding. The fault current will seek out your #10 ground wire to the barn, hop the illicit N-G bond, ride the #10 neutral back to source. That's a mighty ton of current going through those long wires, is either one breakered? No. The long wire run will impede current somewhat, which will make the supply breaker to the failing device slow to trip. It'll spin the electric meter really fast until something fries/breaks/burns.
So the answer is that anywhere neutral and ground arrive in separate wires, "whether to bond them" has only one answer: absolutely never. Except, of course, in the one official location where the equipotential bond must be located.
That shutoff switch
The shutoff switch you bought is probably intended for an air conditioner or motor (hence my suspicion it may have 3 terminals; for 3-phase, to satisfy a rule in factories that every hardwired machine must have a maintenance shutoff switch in direct line of sight to the machine.)
- If it has 3 poles, use one for neutral and use the bus for ground. Leave the chassis bonding screw in.
- If it only has 2 poles, then use them for the hots, use the bar for neutral (remove chassis bonding screw) and then either buy an accessory bar for ground (I see mounting holes) or just wirenut all the grounds together since all the wires are copper. To ground the switch chassis you'll need to find a hole tapped 10-32 and pull a pigtail off it.
Best Answer
Since this is a main panel, neutral and ground can go on the same bar
The reason why you see neutrals and grounds commingled here is because when wiring a main panel, many electricians who wire mostly main panels (vs. wiring subpanels for the most part) find it quicker and simpler to simply slobber all the neutral and ground wires onto whatever bonded bar is handy. They can get away with this because your main panel is the one and only point in your system where neutral and ground are allowed to meet -- they must be kept separate everywhere else.
So, you can simply sort out any spots where neutral and ground have been landed on the same hole, and call it good.
As to that ground screw?
As to the apparently-missing ground screw -- I'd try a Square-D 4028345850K since this appears to be a 100/125A loadcenter. If it doesn't fit into the empty hole on the top right, post back with clear photos of the label on the inside left of the loadcenter cabinet.