If a player is significantly behind in time, does it mean he's bound to lose? Let's focus on FIDE classical chess finals historically.

Caruana is always behind in time control, thinks more, waste more energy.

Statistically speaking, did the players who played faster won more titles? Is Caruana showing signs of weakness?
I don't know about statistics, but I do know that they are not "bound" to lose. It depends on the position. Is it easy to play, or is it a very sharp, tactical position? Also, is there a time-increment? Often, in classical FIDE games, there is. Based on experience, I can tell you that it can be very stressful being low on time in a long game, even if there is an increment. However, if psychological (personal) influences are ignored, it very much depends on the position and on the time-increment.