I find every game lacking in some way. I'm looking for a game with a strong focus on empire building, managing towns and cities and their population, dealing with religion and culture. I'd love a deep military system with troop stats and supply/logistics systems. Maybe also a good political system tying into this, with accomplished generals becoming more powerful militarily and politically over time. I want to feel like I'm conquering savages and civilising the world. Is there anything out there?

Which games disappoint you so far?

I wouldn't say disappointing as such, more lacking in certain aspects. Toral War is too simplistic and Rome 2 especially I find very poorly designed. Old World is excellent, but light on military management and population mechanics, and I find the events somewhat overrated. Far too many "get stuffed, go to war" type events. I'm not sure if I've given FoG: Empires a fair chance: people says it's excellent designed, yet I didn't find myself actually doing much each turn, and I'm really unsure of the randomised buildings mechanic.

Caesar III springs to mind if you haven't tried it although it may not tick every box you are looking for

I was going to say the same, except regarding Caesar II. I don't know if you can go wrong with either, though. Great games.

OP, while it doesn't match the gameplay you described, I'd recommend looking into https://www.gog.com/en/game/a_legionarys_life as it does match "giving a good Roman experience."

Also, while again not the type of gameplay that OP describes, I always like to recommend https://www.gog.com/en/game/the_age_of_decadence . Rome x Post-Apocalypse and, while short and relatively small-scale, it is one of the best RPGs I have ever experienced given my tastes (I like the "roleplaying" aspect + min/maxing and this is a game where the player is essentially locked into certain paths, based on their role. By contrast, think of a game like Skyrim where the player can be the master of everything, the best thief, the toughest warrior, etc all simultaneously. While games like that can be a fun "power fantasy", I find them lacking in role-playing. The Age of Decadence is the opposite; for example, combat for a non-combat character is extremely difficult if not impossible in most instances, as it would realistically be).

I'm not sure whether this is proof of your perfect grasp of the Roman mindset, or desperate need to read up on the ancient world.

Well, going by what you're looking for, AND you're apparently from the UK, maybe what you really should look up, are games that give a good "British Empire" experience, instead? XD

Shadow Empire spoiled you :p

to be fair, there are not many games that really present this wild outside world ready to be explored and conquered.

When Field of Glory Empires released, Paradox also released a title, Imperator Rome. https://www.gog.com/en/game/imperator_rome maybe that title would interest you with its advanced civilization management options ?

Also, i'm really sucker for Total War Warhammer III atm

Why limit yourself to just savages. Civilization 4 lets you conquer other civilizations, and you can play as the Roman Empire if you wish to do so.

https://www.gog.com/en/game/sid_meiers_civilization_iv_the_complete_edition

Try Age of Wonders: Planetfall! You won't be disappointed.

It is a shame the game is not on GoG but.....

Ryse, son of Rome

I think it does a pretty good job of providing us a cinematic Rome experience. You have the crazy emperor, the barbarians that kill your family. Fighting as one with the legionaries. The following campaigns. It is a shame it's a steam only!

How do you feel about map games like Europa?

Alternately, are you much for Koei simulations?

Fields of Glory Empires maybe?
Include Fields of Glory 2 and you got your battles playable too.

And if you want to push the idea of a supply system to the max, Hegemony III.
It is lacking in other parts though.

And on a smaller scale not as a empire building game, we would have Expeditions: Rome

Grand Theft Auto IV gives a pretty good Roman experience.

If you want an authentic Roman experience, then look no further than the Asterix games...