Play on a higher difficulty: Ridiculous or Insane. The game will be much harder - obviously-.

If you can beat the game without exploits on Insane, with the Magic Victory enabled, let me know

Personally, I also prefer to fight the environment above fighting the AI. I don't suggest bigger maps. Maybe more Wildlands. You can select Altar or a custom faction with 'Wanderlust' for more quests. Once you have seen them all, consider buying the Quest Pack or install some mods.

I'd agree that it's very uncommon to win a strategy TBS by surprise.  But some games do a better job of preserving drama in the end game than others.  The Civilization titles by Firaxis always suffered from end game grinding.  So did Stardock's own Galactic Civilizations.

The Heroes of Might & Magic games did a better job of preserving drama through the end game, when you had a titannic collision with your opponents.  Less sandbox-oriented games like Panzer General and the millions of clones it's inspired have you racing a clock to achieve varying levels of victory conditions.  ARPGs like Diablo offered up Boss battles at the end of each "scenario".  There are different ways to preserve a little end game drama, even if the fact that you're going to eventually win becomes a foregone conclusion at some point. 

The old Warlords games didn't make you conquer every city on the map--just a sizable majority--before you were declared victor.  Is there such a Setup option in the FE menus that I'm missing?

In theory, the multiple paths to victory was supposed to provide some drama there.  In practice, I haven't seen it.  But maybe I'm just playing on too low difficulty levels.  I'll start my next game up with a higher AI competence level!

I have not played FA long enough to know all its quirks but there seems to be a sweet spot in the balance of Map size - number of opponents - Difficulty level. 

Large map + high difficulty + few opponents might give you more of that epic feel. I used to play normal maps with 4 opponents. There is almost no wilderness by the time everyone has a few cities. I then tried a large map. Feels much different.

The best games are those that keep you one the line for long.

I remember such an epic GalCiv2 game.  I was sandwiched between the drengin and Dominion of Korx. Both were very strong and korx hated me no end. I was in a constant war with Korx and they had better military than me. I had to shift in overdrive mode and sacrifice my entire production toward massive amounts of warships.

I managed to gain ground on korx but was afraid Drengin would backstab me so I had to pay them tribute and be nice to them and they decided to mop the other nations first.  When I was finally done with Korx I was a battered empire and I now faced a gargantuan Drengin Empire who had just finished conquering pretty much everything else and owned 2/3 of the galaxy ( Me owning the other third)

This ended in a massive 1v1 clash with some overpowered Drengin. I managed to resist the initial onslaught by a clever use of defensive starbase.

That particular game kept me sleepless. This is what I strive in 4x games. But it takes time to find the settings that do it best for you. And its also random, no two games are the same.