For the most part I try and avoid the kind of analytical thinking described in the OP because I know I would absolutely tear the entire story, nay, the entire show concept, to shreds by doing so most of the time. So, I deliberately put my BS-ometer on a very low setting when I watch... but I still think I'm irked far more by some of the nonsense than the average viewer :P
Note, I'm not saying I turn off my critical faculties, nor do I feel I should ever have to to enjoy the show (that would be the time to stop watching it), I just give a very lenient allowance in the logic and science department.
I will say that one absolute necessity to enjoy the show is to treat it as fantasy, NOT real science fiction. It falls some millions of light years below the standard of real science fiction. And that has always been the case, it's not intended as a criticism of the current show.
BUT, one thing I cannot stand is the creators of the show taking too much license from the above philosophy and feeling they can make the show nonsensical. That has happened a few times in recent series: 'Kill the Moon', 'Forest of the Night', 'Robots of Sherwood', the whole Viking and Zygon things last season, which I cannot rewatch and genuinely eroded my affection for the show a little (first time I've considered not watching the following week since I 'rejoined' the show in Season 5). As long as the writers are conscientious I give them a lot of leeway. If they simply dash off rubbish then its a different matter.
But simply start thinking about any of the most basic aspects of the show (everyone in the entirety of space-time speaks English for example... the TARDIS translating is NOT a credible explanation because it's ability to do so basically amounts to magic, unless we are to believe every language to have ever existed anywhere is in it's databanks. 'Translation devices' are one of the worst plot devices in all science fiction for this reason) and it falls apart... so you have to just give it that allowance if you value the strengths of the show enough to permit it those weaknesses.