Lance Odyssey is probably comparable to Conquest in effectiveness. Non-beam Odyssey is probably about as effective as a cruiser, which is not good enough for how much it eats.
I do not like the current Odyssey design. There is one true loadout, lance sniper, which is very hard to assemble (due to weapon and hullmod rarity), while the rest (including classic loadouts that used to be viable) are traps. Loadout of lance Odyssey is so rigid that it might as well have nearly everything (aside from which hullmod among Expanded Deck Crew and Hardened Subsystems and second bomber wing you need for a particular fight) builtin.
If Lance Odyssey will be the official Odyssey, it needs to be redesigned to that it does not have so many empty mounts (three missile mounts which will never be filled and at least one small mount due to lack of OP). It also needs to widen the sweet spot because Odyssey really needs to focus-fire three lances. (AI might not have much problem aiming three lances, but then again, AI lance Odyssey can only fight well in one-on-one duels. Its attacks are too unfocused against a crowd.)
It is probably better if lance Odyssey was split into a different ship, a new high-tech capital that does everything Lance Odyssey can do, without the wasted mounts and weird turret arcs, maybe less peak performance too since its fighting style consists of outspeeding and outranging everything. I do not know if people would enjoy fighting against it controlled by enemy AI, since such a ship can run away like a Timid ship. Currently, there is no way to fight against triple lance Odyssey in a campaign without mods. Meanwhile, classic-style Odyssey should get significantly buffed, but turret arcs changed so that triple lance sniper is no longer possible (because it is so much better than every other Odyssey loadout).
If classic-style Odyssey with terrible shot range is to be used, it needs to have much better shields, perhaps 0.6, and maybe better flux stats, so it can approach enemies safely and slug it out. Without lances, Odyssey has no choice but to slug it out like an Aurora, but since Odyssey is capital-sized and is too big and slow to hit-and-run with short-ranged weapons, it needs to be tanky enough to fight like a capital. Or perhaps instead of huge durability buff, its system can be replaced by Phase Skimmer (which works wonders at dodging stuff) and durability might not be raised as much.
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Paragon had a harder time initially until it made it to the wall. (In a campaign fight, Paragon would not need to go so far into the open to attract enemies.) Once Paragon made it to the wall, it absolutely destroyed the enemy, and relatively quickly. Admittedly, it used much of the same rare stuff Odyssey needs. Onslaught had a harder time, but it did not need rare stuff beyond ITU (which will eventually drop after enough endgame fights).
Odyssey had to run for its life from frigates much and it needs time to pick off isolated targets. Odyssey could avoid getting surrounded, but it did not have infinite time to kill things. Peak performance was the greatest enemy. Odyssey had to take some risks to kill ships before CR timed out. Conquest fought similarly as lance Odyssey.
Against smaller targets, Odyssey did not necessarily need hard flux. With three lances and four lasers, soft flux was overwhelming enough to destroy frigates and some destroyers. It is only against bigger ships that soft flux is very important.
The point of solo simulator fights is an approximation of campaign fights, albeit without campaign skills on enemies that can turn otherwise easy fights much harder.
I have had one-on-one fights against capitals in the campaign. Sometimes, my fleet would take too many casualties if they engaged with the capital, so I send them elsewhere to fight everyone else while my flagship duels the enemy capital.
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Apogee is a bad ship because it lacks the speed and the tools to fight effectively. It is too slow, and it does not have the shot range to engage effectively without getting surrounded. Apogee has noticeably better capacity than other cruisers, which makes it more like a high tech Mule, but Apogee comes too late when hybrids are useful. It might have been alright if it was easily accessible and found early in the game like Mule or Venture, but it comes too late. As a warship, it is sub-par.
I do not know what other campaign advantages Apogee bring. Apogee's builtin surveying equipment is eventually available to everyone (that gets put on all of the freighters and tankers if I bother with surveying), and high resolution sensors eventually does not matter when I want enemies to find me and fight. Anything useful Apogee could bring for non-combat, I am better off bringing a dedicated hauler.
Odyssey can haul a bit more supplies than other capitals, but not enough to matter. That is like saying Wolf is a good enough hauler to qualify as a hybrid (which it does not) because it can take ten or so more supplies than Lasher. Odyssey used to have decent fuel capacity before 0.8, but that probably got whacked when fuel was gutted for every ship that was not a tanker, not to mention Odyssey guzzles ten fuel per light-year. If anything, Odyssey is more similar to either Aurora (classic loadouts) or Conquest (lance sniper). Modern Odyssey is a dedicated warship.
As for fleet, that is not a point in Odyssey's favor (as a flagship) because it is not a front-line unit, not to mention ships in general benefit from a fleet. After all, a ship that can help itself and others is better than a ship that requires help. Currently, carrier fleets are probably the most effective fleet in the game. Due to AI faults, AI carriers love to send their fighters (if you want heavy fighter spam) to escort your flagship, and if you want to use their fighter escorts (who insist on buzzing around your flagship like flies) to attack ships, you need to wade in and tank for your carriers, and Odyssey cannot do that as well as others (because it is too fragile and too slow to escape if it gets caught). Your flagship needs to be somewhat close before your fighters decide to attack whatever you are attacking. (Astral flagship has a terrible day, it must tank for the other carriers just so all fighters from your fleet attack something instead of hovering around your flagship.)