Does power distribution within a house need individual neutral lines from the circuit breakers?
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 &quot;rule&quot;: 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 &quot;on&quot; part of the circuit (the other hot) returning through the shared neutral and zapping the person working with the &quot;off&quot; 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 &amp; 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?.