There are three problems that I have with this:
1. It requires that battles take sufficiently long that fleets on the far side of the sector can reach the battlefield before the fight is over. However, fighting a battle and winning it take no more of a ship's (or of a fleet's) movement points than a standard move action does, which implies that the actual time taken for a battle is no greater than (1 week)/(how fast the ship is, neglecting units). This would imply that no ship which is further away than that fraction of its movement points can possibly reach the battlefield in time to affect the outcome, though they could disturb the victors. If we assume that all factions keep pace with one another in terms of engine technology and average fleet speed, this basically works out such that only the fleets in tiles adjacent to the attacked fleet can reach the battlefield in time to affect the outcome (this is still an improvement on the GCII style, however, as it at least still permits you to engage multiple fleets at once). Additionally, I don't think I normally made ships which are even capable of traversing a full sector per turn in most of my games, even though I play on immense maps, because I don't usually feel that that kind of speed is necessary, or even particularly beneficial, because it takes too much space on my ships, yet I often reached the end of both the ship construction and the engine technology trees in my games, and I normally played on immense maps. Speeds of 10 or 12 parsecs per week, yes, those speeds were something I usually attained; speeds of 15+ parsecs per week, though? I'm fairly certain that one's a no, except on special-purpose hunter/killer groups or colonizers/transports/constructors/scouts/other non-combatants.
2. Ships which respond to engagements in this fashion should lose movement points in their next turn to compensate for them running over X parsecs in order to reach the battle (they would not necessarily lose X movement points if we take the listed speed of the ships to be a normal cruising speed with space leftover for emergency use, but emergency speeds aren't standard speeds because they're generally fuel-inefficient and taxing on the engines, and doing this would likely mean that the ship misses most or all of any scheduled resupply rendervous; all of this would impact its performance in the subsequent turn, and there's also no in-game evidence that the engine speeds we are given in the game do not represent the maximum possible speed attainable by that a ship with a given engine configuration from a specific world of origin).
3. If all engagements always force all ships in the affected sector to respond to the battle, or if the AI always makes all its ships respond to the battle, then I've suddenly obtained an easy way to lock down every ship in a sector simply by sacrificing a couple of cheap ships every turn; this is doubly annoying if the AI can do this to me. Presumably all the responding ships would end up in the vicinity of the initial engagement, which will greatly slow their trip to wherever they were supposed to be going unless they were supposed to be moving in the direction that the engagement pulled them, which is only particularly likely if there is only one primary direction in which valid targets lie for the AI or for me to be sending ships to attack and no reason for me or the AI to be sending ships along paths off this axis to, say, reinforce a recently-weakened sector or replace a fleet in a now undefended sector (i.e. launch a counter-attack before the troopships can secure the sector). Even if responding to the engagement doesn't cost the ships movement points in the next turn, it should still bring them to the battle site, because quite frankly an entire sector is far too large of a battle space to be practical even with roughly endgame GCII ship speeds (up to 21/26/36/41/46 parsecs per week on Tiny/Small/Medium/Large or Cargo/Huge hulls using the Terran tech tree and not accounting for race or planetary speed boosters, on ships maxed out with only engines; notably, even the Huge vessels would still take about one-third of a week to traverse the 15 parsecs of a sector, unless we're assuming that the ships spend most of their time sitting still in space, which would also imply that there's even less time to respond to an engagement).
That said, I would not mind if either fleets on neighboring tiles or the stacks of fleets on any given tile could all engage at once, possibly with a delay before the non-initiating fleets join the fight so as to give some time for the initial battle to be wrapped up in one side or the other's favor. It would certainly help with the late-game clean-up phase, it would help with the issues of planetary assault when a faction stacks lots of ships there to take advantage of the Orbital Command Center's bonus to planetary defense, which ignores logistics limits, it would allow me to contribute to the battles my allies engage in during common wars, and it would allow a better chance at protecting weakened fleets in heavily contested space.