I like this idea. And I think a very elegant solution to make this possible is by linking the likelyhood of every crisis to the legacy points claimed during the age.
So if the players (human and AI) claimed a lof of economic legacy points, the plague will be more likely.
In the end there could be one crisis associated to each of the legacy path with the chance of this crisis increasing with every point earned in this path by any player.
I posted something, a thought experiment really, about linking Crisis directly to Legacies, but I'm not so sure about that now.
Any direct link makes it far too easy for the human gamer to 'select' the type of crisis he wants - or the crisis he knows will most handicap the AI - and make the human gamer Snowball even worse. I believe the whole idea of the crisis and transition to new Civs was to keep the human gamer from running away with the game half-way through - which was far too common in Civ V and Civ VI.
They still haven't managed that very well, as numerous posts on this Forum indicate, but that doesn't make forcing the human to rethink his game not worth pursuing, and I'm afraid giving him a direct path to selecting the type of Crisis will do just the opposite.
One possibility that I'm mulling now is to make the type of Crisis and duration and degree of severity dependent not only on the human player's actions, but even more on the AI Civs' and the IP actions.
As in your example, a Plague Crisis would depend on the economic legacy points claimed by All the surviving players, or the number of active Trade Routes in all the surviving civilizations. That way, the human player (or players) would have much less control over 'selecting' the crisis.
And adding to this, there should be, especially at higher difficulty levels, a very good chance of getting more than one Crisis. After all, Rome's IRL crisis started with multiple Plagues (2nd century) followed by multiple 'barbarian migrations/invasions' (3rd- 4th centuries), followed by an on-going Loyalty crisis as every would-be emperor/general tried for the throne (4th - 5th centuries).
And, following the historical example, the gamer should have the potential option of 'muddling through' all the crises with a possibly much-weakened Rome (historically, Rome never disappeared, it just dropped from a megapolis of 1,000,000 peole to 30,000: catastrophic but not entirely apocalyptic), or transitioning to a near-identical Byzantium or throwing up his hands and becoming Normans, Burgundians or Goths (the latter looking, perhaps, suspiciously like Tang Chinese) and effectively starting over with mostly new Uniques and Legacies. Note that if designed properly, each of those choices would have viable consequences both good and bad, but in each case very different.
In my view, Gamer Choice is the key to a better game.