I still never get the "coincidence" argument for why a movie is bad. I always wonder if people have the same criticism for a novel like Great Expectations.
I don't know Great Expectations.
A plot with too many coincidences is bad because it's not believable. Lack of believability means lack of immersion and, if it's too much, lack of me caring. Real life is not an alignment of 1/10000 odds that all come true. It also makes the plot predictable.
The fact is that extremely unlikely events occur all the time. If I ask my computer for a random number between 1 and a million, and it gives me 447,423, well, there was only a 1 in a million chance that that would occur. But no matter what number it gives me, there was only a 1 in a million chance of it occurring.
That's missing the point.
You can explain every event away by listing a number of possible outcomes that all have equal probability and saying 'all outcomes have been equally likely!'. For example, if you buy a lottery ticket, you could say that it's not particularly unlikely that you got everything right, because the set of numbers that were selected has been exactly as likely as any other set.
What matters for a judgement of 'likely' or 'unlikely' is how you grouped/would group your sets of probabilities before the event happens. If you buy a lottery ticket, you'll group them in 'loss' and 'win' and 'win' is a lot less likely, so if it happens, it's an unlikely coincidence, whereas 'loss' is really likely, and you don't really care how exactly the loss came to pass.
Similarly, in your number example, if you generate 447423, that's not an unlikely coincidence. If you had gone up to it saying 'I want it to generate the number 447423 and then it does it, that would be an unlikely coincidence. Similarly, if you draw 4 cards and they're all aces, that's an unlikely coincidence, because you perceive aces as something special. If you draw 4, 7, J, J in three different colors, that's not an unlikely coincidence, because you had just grouped it in with a ton of other outcomes as 'nothing special'.