See, the fact that the screwdriver's effect is expedient is what makes it matter. For instance, if a monster's chasing the Doctor, there's suspense. It's breathing down his neck. He doesn't have time to pick a lock. BZZZT! Open door. BZZZT! Seal door. Monster escaped.
If someone's captured, restrained, and in some kind of peril, there's tension. Only moments to act. How can the Doctor possibly rescue them with the bad guy right there? BZZZT! Restraints gone. Victim saved. Turn attention to foe. He can even due this to non-mechanical stuff like webbing.
Fair enough; I guess everyone's "line" is different. The sonic screwdriver falls below the line for me - I just find it a convenient narrative device to move the story along rather than have him circumvent a computer for 10 minutes or pick a lock or what-have-you. But I can appreciate that it may bug other people!

Of course, one should bear in mind that you are watching a kids' TV show. The target audience is 12 years old.
P.S. I keep thinking Bill Murray whenever I see that picture you're using.
That's a picture of me drawn by Claudio Pozas.

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