I think Doomed is an amazingly cool weakness, and those who feel like it denies player agency might want to take a closer look at the math. I ran some simulations, and assuming I did it right the numbers look very fun. In these simulations, I ignore the possibility of drawing Doomed or its replacements twice during one scenario, since that should be uncommon (but it would be quite bad).
Let's say you run 8 scenarios, and draw half your deck each time; specifically, let's take a 28-card deck (standard 33 minus the starting 5) and draw 14 cards per game. In 8 scenarios, Doomed will kill you 36.4% of the time and there's nothing you can do about it.
Or is there? What if you try to draw fewer cards (by pushing to complete faster, taking fewer draw actions, etc)? Just by reducing by 1 card draw per game, and taking 13 instead of 14, your odds of death by doom drop to 28.8%. Reduce to 12 cards and it's down to 22.1%. Alternatively, suppose your deck is built for card draw and you ignore Doom and pull 16 cards per game: you'll die 52.7% of the time.
Maybe the worst happens and you draw Doomed the very first scenario. Your odds of death (back to 14 cards here) are now 50.0%. Maybe you play the remaining games super aggressively, deciding that the risk of defeat and trauma during a scenario isn't that big of a deal now, and only draw 10 cards per game: with that strategy you can push your doom risk back down to 21.2%.
If you have better luck and survive the first scenario without any Doom, your (14-card) odds are down to 22.7%, so you aren't out of the woods yet. Survive 2 games sans Doom and you can breathe more easily: 10.9%. Draw Doom once in the first 2 games and you're at 34.4%, close to where you started... and even with only 6 scenarios left, the impact of drawing 1 fewer card per game is still significant.
Note that, because of the way the probabilities work, drawing Doomed/replacements 4 times over the course of a campaign is much more likely, and many players will find themselves with The Bell Tolls in their decks by the final scenario (something like 50% using base assumptions). Also I did not include the effect of adding additional story assets or weaknesses over the course of a campaign, but it's fairly significant too, so whoever has Doomed should try to pick those up when possible.
Bottom line is Doomed makes a situation similar to the chaos bag; players can change the odds at a cost, but aren't in control of the outcome. The interesting difference is that the cost/risk picture is very long-term: each avoidable card draw hurts you very little (probably), but over the course of the campaign the likely cost for making a lot of avoidable draws is enormous. It's very much in keeping with the concepts of the game both thematically and mechanically.