Hi Dragonstone,
It all depends on what you call "tsumego". Practicing problems is indeed an important part of the training.
But playing games is an important part too...
As well as studying the fundamentals (haengma, strategy, sente, jôseki...)
...and from time to time, having one of your games reviewed.
So you are right to play games. A lot of games ! It is as important as practicing tsumego, if not more.
Exercises should include the practice of capturing techniques (nets, ladders, snapbacks, throw-ins, pins, jachung...), the practice of life and death (vital points, false eyes, capturing races...). All these subjects should be included in "tsumego".
But exercices should also include tesuji for connecting, for escaping, for enclosing the opponent, and also the study of haengma (cross-cut, horse move for attack, one-point jump to escape, how to deal with races on the 2nd line, on the 3rd line...), the endgame (double sente moves, reverse sente moves, monkey jump, endgame tesuji...).
And these are only the tactical side of the fundamentals principles of go. The strategy is the other part : assessing a global position, spotting weak groups, extending from corners, how to use a moyo, choosing between reduction and invasion... There are very few exercices about these very important subjects.
Doing just tsumego with "Black to play and kill" is very restrictive.
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and these few game situations were normal and easy tsumegos.