This is totally nonsense. You are just listing certain bonuses of a civ and asserting because of that they don't need redemption. That might make sense in a game where all the civs are reasonably balanced. We know that AoC isn't one of those games - there are about 3-4 viable civs on any given map for 1v1, and a pool of 6-7 for TG.
No, I listed reasons why they'd be too powerful if they had a strong monk line. The Goths already have cheap infantry that counter cavalry and archers and cost effectively beat other infantry. If they also had monks with redemption, then they would counter siege as well. If you think pikemen are too easy to back up monks and mangonels, then Gothic infantry are valid reason to give them weaker monks.
The French meanwhile have free farming upgrades, "free bloodlines" and cheaper castles. That means by the start of the castle age they've already saved a lot of resources. You said a castle is the only proper counter to an Aztec monk rush, and in this case they put a stop to weaker monk civilisations with a castle that costs 162 stone less than other civilisations. Given that, French redemption would allow monks to push forward, driving back siege, while villagers put down a cheap castle, all while expanding their economy with 375 food farms. That's also reason to give them weaker monks.
The Byzantines have cheap skirms, cheap pikes, cheap camels, strong walls and buildings, cheap imperial age -> they OBVIOUSLY don't need to convert mangonels, yet they are one of the most prolific monk civs. The Chinese have cheaper upgrades, a faster start (contrary to popular belief) , a strong tech tree, better farms, I doubt they need to be able to convert mangonels. The Spanish build quickly an thus have a better start than average, have the strongest UU in the castle age, BS upgrades don't cost gold: I don't think they need to be able to convert mangonels. The bonuses of these 3 civs are far better than those of Franks and Goths.
The Byzantines are a defensive civilisation without economic bonuses. Although they have cheaper counter units (and a cheaper imperial age), they are not very fast at pushing. Although they are able to take full advantage of all these bonuses on arena, this is not the only map in the game. On maps like arabia and nomad, the strength of their monks is very much balanced out by their lack of economic bonus.
The Chinese may have a strong economy, but in the late game, they have neither heavy cavalry nor heavy siege. They need to push home their advantage in the mid-game. If they didn't have good monks, then on closed maps, it would be too easy to wall and boom against them until the late game, where their weakness to onagers begins to show. Similarly with the Japanese, who have slightly stronger siege, but weaker cavalry.
Pikemen are really cheap. 8 pikes, 6 monks with santitiy/redemption, and 3 mangonels are already very difficult to stop for any civ without redemption. I did a similar strategy in a BF game the other day and was able to imp before 32 min with 100+ vills. That being said, my opponent did not play very well due to bad start/bad reaction, but that was only accentuated by the sheer difficulty in pushing that combination back.
Was it a non-mirror match-up? Because the right response there does depend on what the civilisations are. Independent of civilisations, the most consistent response is early detection. It's when they start driving you off vital economic zones that it becomes cost effective for them, and that's normally when they hit suddenly. If you manage to get a castle up before you've lost too many farms, then their monk rush can turn into a bit of a waste.