Does power distribution within a house need individual neutral lines from the circuit breakers?

Actually, house cabling is done in a "Tree Topology" not a "Star". A star is a kind of tree... of course, so is a vine. The point is your cable routing can branch at any point you want, but it can't re-join another branch, and the wires especially can't create a loop.

Safety ground excepted, because it is not a current-carrying conductor.

Neutral is not ground. Seriously.

Current flows in loops. Neutral is the normal current return path. Really, neutral is just another hot conductor, except the earth bonding was chosen in a manner that happens to put neutral fairly near earth, if the Grounding Electrode System is working properly. In fact I had a building where that was not the case at all, and neutral was 120V from earth, and it all worked fine.

So on a simple circuit, half your current flow is on neutral; it is equal and opposite to flow on hot.

Current never flows on equipment safety ground except during a fault condition. It sinks wayward lighting and ESD to earth, and human-made current back to source. We hope that induces an overcurrent or GFCI trip.

It causes all manner of trouble if one is used in place of the other. In fact doing so is worse than running ungrounded: It's normal that if the neutral wire breaks, it floats up to line voltage; that's why we insulate it. But if some fool has bridged neutral and ground, that means grounds are now electrified too!!!

Why does a transformer work on AC but not DC?

Suppose you run two wires out to a point-of-use: one supply, one return: current is making a loop on these two wires. The wires are parallel and tightly packed together. Does each wire throw an Electro-Magnetic Field? Yes. Are they opposite since currents are opposite? Yes. Since it's a loop, currents are equal and opposite. Does that mean the EMFs cancel each other out? Yes.

Suppose you don't do that. You take a circuit in which current is flowing, and wrap their wires so the current flows in a loop around something. What happens?

Well, we know in DC power, it sets up a magnetic field. But if the DC current is non-changing, it's just non-moving - no different from a refrigerator magnet.

But what about alternating current? Uh-oh. Now you have a very animated magnetic field. Pity any nearby compass; it's going to be jinking back and forth at line frequency. So anything ferrous is going to be vibrating in that field.

What will it be pushing against? The wire itself. So the wire will be vibrating too. What could go wrong? Now we're into the world of metal fatigue, and then, wire overheating due to reduced cross-section due to cracking, and then, arc faults. What great firestarters those are!

We also know that's how the inside of a transformer works: if there was another loop of wire in the vicinity, we'd be inducing AC power onto it.

Now, why are transformers laminated? To break a type of magnetic flow called "eddy currents". What happens with eddy currents? At the least, eddy current heating.

Now, an induction motor rotor is just an aluminum cylinder. Why is it slotted? Again to cut those eddy currents and force magnetism to run the length of the rotor. So what if it's not slotted? Again, eddy current heating. In non-ferrous metals. Really.

So the upshot of this is, Having power take a "world tour" around your house is a bad idea. In fact it's prohibited; look at NEC 300.3 et.seq.

Neutrals don't have breakers

The entire concept of neutral not having breakers is because neutral should never be handling current for anything except its partner hot. (or hots in the case of an MWBC).

So as soon as you glob all neutrals together into a web, you now have a big problem. One neutral could return more than its capacity, and you'd never know it.

In fact, a neutral wire breaking is perfectly routine and unsurprising event. In proper wiring, what happens is simple: the circuit quits working. Person goes to fix it, hot is working, asks us here, we tell them to check neutrals, voilà. Your system would keep on working, with neutral detouring onto other neutrals that don't have the capacity for it and their normal traffic. You'd never know if you were overloading neutral, unless you double the size and expense of your panels to fuse neutral.

You must stop all neutral current to maintain the system

Routine maintenance often requires severing the neutral wires temporarily. You can't do that if there's current flowing through the neutral, because severing will instantly cause a voltage differential equal to line voltage. So you need a practical way to de-energize the neutral wire (or to be more precise, all the hots feeding the neutral). Your system provides no way to do that short of shutting off the whole service, which means working in the dark without power tools. NEC and local jurisdictions do not allow that.

"I don't need GFCI" is bulloney

Your logic is that you're going to rewire this some time in the future. Not without GFCI/AFCI you are not. AFCI became a requirement for some circuits in NEC 2005. And by 2014 virtually every circuit required one or the other, and in 2020 some require both. AFCI has a built-in "mini GFCI" as that is cheap way to detect H-G and N-G parallel arc faults. Neither one works with shared neutrals. And, you may not know this detail, but AFCI can't be done at the recep, it must be at the breaker.

Most of the 120V world uses El NEC. Canada basically copies it and gives it a different name. Japan and the Philippines pay close attention to NEC changes. (you won't be doing this in the Philippines anyway; both their wires are hot). Everybody else goes off the European playbook and they require whole-house RCD: you might evade an RCD trip there, but it'll still be a Code violation at several levels.

Fascinating question. Definitely not to code, but conceptually:

• Neutral carries power. On a 120V (US) circuit, neutral carries as much power as hot. So that gives rise to the first "rule": Neutral wires must be sized to match the hot wires. That normally means: 14 AWG (or larger) for both hot and neutral on a 15A circuit, 12 AWG (or larger) for both hot and neutral on a 20A circuit, etc. In theory, multiple circuits could therefore share a neutral, provided the one neutral was large enough for all the hots put together. So if you have 2 x 15A then you could have 2 separate 14 AWG hots and 1 combined 10 AWG neutral. The neutral size would rapidly increase as you add more circuits - to the point where you would need a neutral the same size as your service entrance neutral snaking throughout the house. That would be a tough wire to run (thick and hard to bend) and not cheap.
• There is actually a combination already in use: Multi Wire Branch Circuit. An MWBC uses one neutral for two hots. But the key is that the two hots are on opposite legs, so the neutral actually gets the difference instead of the sum. A 15A MWBC maxed out will have 15A on each hot and no current at all on the neutral! That means the same 14 AWG can be used for the hots and the neutral.
• Neutral must never break. In an ordinary circuit, that is no big deal. The other failsafes (grounding, GFCI, other design aspects) mean that if the neutral breaks (but not the hot) then your risk is generally minimal. But that risk is enough that an MWBC must have the neutral connected in a way that working on one part of the circuit won't affect the neutral for the rest of the circuit. If you have one giant neutral for everything, that could get really messy.
• Common Maintenance Shutoff - An MWBC must be set up so that if part of the circuit is turned off for maintenance, the entire circuit is shut off. That is to prevent power on the "on" part of the circuit (the other hot) returning through the shared neutral and zapping the person working with the "off" part of the circuit. A whole-house shared neutral would therefore mean everything off any time you work on any circuit.
• Magnetism, heating, etc. - I honestly don't understand this part. But basically code generally requires that all wires in a circuit be together and have (nominally) equal current running through them, or be widely separated (e.g., some old knob & tube). Otherwise some funny things can happen that can be major fire hazards. A whole house shared neutral would have this problem unless it was bundled with all wires everywhere - which of course would more than balance any savings of the shared neutral. @Harper has a detailed explanation of this item in Why does a transformer work on AC but not DC?.