Why the fuck does anyone watch the show? Hell, how is it even still on the air? Dude apparently has at least two god-tier items of Plot Device-itude that can do literally anything, not to mention he's immortal and a time traveler. There is no potential for drama unless the characters suffer ridiculous amounts of amnesia and forget all the crap they could do to solve any problem.
For the god-tier items they tend to do a number of things to take them out of the picture:
1)Many times the TARDIS is disabled/removed from the equation at the start of an episode. Acid traps it underground, a rock slide covers it, something related to the problem makes the area it's in deadly, it gets carried off by aliens, it's self-activating safety mechanism carries it away, you get the picture. The few time where this hasn't happened and it's not used are generally because A) It's on the other side of the city and they need to chase something
right now, or B) whatever they are up against has enough power to trap it, and they don't want that to happen (examples are when he is facing the Master/the Daleks).
2)They specifically set up things that the sonic screwdriver can't solve. According to current canon the sonic screwdriver can't be used to "wound, maim or kill living things". This makes it fairly ineffective at stopping a rampaging werewolf or other biological creature, which constitute the majority of the Doctor's one-off enemies. In fact pretty much the only times it has been used offensively was to either cause a brief disorientation to allow an escape (when facing the Silence) or when he had multiple sonic devices and the interference caused by use of them simultaneously made peoples' ears hurt.
3)As for those few enemies/devices that the sonic screwdriver should be able to solve, they are either advanced enough to take countermeasures (the Cybermen or the Daleks) or for devices they simply apply a bit of magical plot "deadlocks" to the device and the sonic doesn't work there either.
As for the immortal/time traveler thing, there are a few things they've thrown into the canon to stop that from being too OP.
1)He's not actually immortal. Unless they produce a way to add in more regenerations (which has been shown to be possible with the Master), he's only got 2 more after the current one (12 regenerations = 13 total lives - 11th doctor = 2 left).
2)Whoever he currently is still "dies" when he regenerates, losing their personality and their emotional attachment to any memories. That lets them pull death drama out of a character that isn't technically dying.
3)He's not allowed to "cross his own timeline" and only does so in times of great peril. This stops him from using the typical time traveler solutions of going back to 5 years ago and stopping his enemy before they become powerful (It's the whole GURPS
Time Travel supplement rules, once he's witnessed it then it's fixed, so he can only fix the problems
now instead of of being able to fix them in the past).
TL;DR: They manage to carefully balance the god-tier strengths with some powerful counters to maintain a semi-stable, drama producing, show. It's like superman and kryptonite. You never see a good superman villain without them invoking kryptonite at some point, and you never see a Doctor who enemy without some sort of limits/counters to him.
And personally I think in terms of combat a jedi would totally beat Doctor Who. Partially because their tool is actually a weapon, and they have intrinsic magic powers where the Doctor only has a clever use of technology. If we gave the Doctor access to his technology then he would probably win though. It's sort of like Batman, give him enough time to think and access to his tools and he can beat just about anyone, but take away everything but his body and the people who have intrinsic super-powers will wipe the floor with him.
Edit: Dang you people for posting so much! You are forcing me to have long-ass posts that nobody wants to take the time to read completely!
**Something they do with alarming frequency. Well, yeah, something you can expect from a millenia old craft flown by an incompetent pilot.
It's implied that there is a lot of traveling going on that is just simple sightseeing that happens between episodes. Therefore my theory is that for every crazy even that happens where the shields end up going down, there are probably dozens more where the shields don't go down and nothing interesting happens that occur between episodes.
I don't even think he drives it. At one point it manifests itself in the form of a lady and it's revealed that he just hits the switch, having no goddamn idea where he's going.
I think.
Same sort of thing here. Since the show only shows us the exciting episodes, there are plenty of times where he gets exactly where he wants to go and everything goes as planned between episodes. We just only get to see the few exciting times when the TARDIS takes him somewhere that he
needs to be rather then where he
wants to be.