Why isn't the sum of Precision and Recall a worthy measure? What is the best way to explain why $\text{Precision} + \text{Recall}$ is not a good measure, say, compared to F1?
 A: The short answer is: you would not expect the summing of two percentages which have two different denominators to have any particular meaning. Hence, the approach to take an average measure such as F1, F2 or F0.5. The latter retain at least the property of a percentage. What about their meaning though?
The beauty of Precision and Recall as separate measures is their ease of interpretation and the fact that they can be easily confronted with the model's business objectives. Precision measures the percentage of true positives out of the cases classified as positive by the model. Recall measures the percentage of true positives found by the model out of all the true cases. For many problems, you will have to choose between optimizing either Precision or Recall.
Any average measure looses the above interpretation and boils down to which measure you prefer most. F1 means either you don't know whether you prefer Recall or Precision, or you attach equal weight to each of them. If you consider Recall more important than Precision, then you should also allocate a higher weight to it in the average calculation (e.g F2), and vice versa (e.g F0.5).
A: Adding the two is a bad measure. You'll get a score of at least 1 if you flag everything as positive, since that's a 100% recall by definition. And you'll get a little precision bump on top of that. The geometric mean used in F1 emphasizes the weak link, since it is multiplicative; you have to at least do okay with both precision and recall to get a decent F1 score.
A: F1 score is especially valuable in case of severely asymmetric probabilities.
Consider the following example: we test for a rare but dangerous illness. Let's assume that in a city of 1.000.000 people only 100 are infected.
Test A detects all these 100 positives. However, it also has 50% false positive rate: it erroneously shows another 500.000 people to be ill.
Meanwhile, test B misses 10% of the infected, but gives only 1.000 false positives (0.1% false positive rate)
Let's calculate the scores. For test A, precision will be effectively 0; recall will be exactly 1. For test B, precision will still be rather small, about 0.01. Recall will be equal to 0.9.
If we naively sum or take arithmetic mean of precision and recall, this will give 1 (0.5) for test A and 0.91 (0.455) for test B. So, test A would seem marginally better.
However, if we look from a practical perspective, test A is worthless: if a person is tested positive, his chance to be truly ill is 1 in 50.000! Test B has more practical significance: you may take 1.100 people to the hospital and observe them closely. This is accurately reflected by F1 score: for test A it will be close to 0.0002, for test B: (0.01 * 0.9) / (0.01 + 0.9) = 0.0098, which is still rather poor, but about 50 times better.
This match between score value and practical significance is what makes F1 score valuable.
A: In general, maximizing the geometric mean emphasizes the values being similar. For example, take two models: the first has (precision, recall) = (0.8, 0.8) and the second has (precision, recall) = (0.6, 1.0). Using the algebraic mean, both models would be equivalent. Using the geometric mean, the first model is better because it doesn't trade precision for recall.
A: It's not that $\text{Precision} + \text{Recall}$ is a bad measure per se, its just that, on its own, the resulting number doesn't represent anything meaningful. You are on the right track though... what we are looking for is a combined, average of the two performance measures since we don't want to have to choose between them.
Recall that precision and recall are defined as:
$$\text{Precision} = \frac{\text{True Positive}}{\text{Predicted Positive}}$$
$$\text{Recall} = \frac{\text{True Positive}}{\text{Actual Positive}}$$
Since they both have different denominators, adding them together results in something like this: $$\frac{\text{True Positive}\left(\text{Predicted Positive}+\text{Actual Positive}\right)}{\text{Predicted Positive}\times \text{Actual Positive}}$$
... which isn't particularly useful.
Lets go back to adding them together, and make a tweak: multiply them by $\frac{1}{2}$ so that they are the stay in the correct scale, $[0-1]$. This is taking the familiar average of them.
$$
\frac{1}{2} \times \left( \frac{\text{True Positive}}{\text{Predicted Positive}} + \frac{\text{True Positive}}{\text{Actual Positive}} \right)
$$
So, we have two quantities, which have the same numerator, but different denominators and we would like to take the average of them. What do we do? Well we could flip them over, take their inverse. Then you could add them together. So they are "right side up", you take the inverse again.
This process of inverting, and then inverting again turns a "regular" mean into a harmonic mean. It just so happens that the harmonic mean of precision and recall is the F1-statistic. The harmonic mean is generally used instead of the standard arithmetic mean when dealing with rates, as we doing are here.
In the end, the F1-statistic is just the average of precision and recall, and you use it because you don't want to choose one or the other to evaluate the model's performance.
