Electronic – Advantages on using a function generator against a multi-pulse generator by using a 555

Not a complete answer, but to address one of your points:

I am thinking that for generating square waves it can be enough to make a pin output high and low on desired time frames.

That depends on your application. This is considered "naive" square wave synthesis, and doesn't produce a mathematically correct square wave. (It's equivalent to sampling an ideal mathematical function without putting it through an anti-aliasing filter first.)

This also applies to triangle waves, sawtooth waves, and anything else with harmonics above the Nyquist frequency.

It will often be "good enough" if you have many samples (or time frames) per cycle, but not otherwise. For example, if you generate a 10 kHz square wave with a 44.1 kHz sampling rate, it will look like this:

You can see that every few cycles are different lengths. The transitions can only occur on sample boundaries, but an actual square wave would transition at a time in between them. Practically, this results in lots of aliased harmonics below the square wave frequency, which you probably don't want, depending on your application. In audio applications, this sounds awful.

You can avoid this by generating a correct band-limited square wave in software, or by using a sampling frequency much higher than necessary for your signal.

Here's a comparison of the two methods on a 5 kHz square:

Simplistic:

Mathematically correct (generated with additive synthesis):

Disconnect pin 6 (threshold) from the output and pull it to ground with a resistor. You can still pull it to supply voltage to manually reset the output.

Note that the output will go to the full supply voltage instead of 2/3 of it as in the given circuit.