Is this how electrical engineers design circuits with transistor switches in them, is this the proper method to calculate the values or does this one have some downsides to it?

For basic saturated switching, yes, this is the way it's done.

Although from the postings in this thread, you've seen it can be done actually 3 different ways

1) using just the test values from datasheets, i.e. Vce(sat) usually shown at a few different test currents

2) using the charts (find the collector current you need, scan across the chart to get the base current needed)

3) using the Hfe(min) or Hfe(typ) values. Some people use the typical values, others prefer the minimum values.

then adding 2x to 3x more base current than actually calculated, "just to be safe" and to cover all the variability in manufacturing or sourcing equivalent parts from different manufacturers.

As for the ohms law question, calculating the base resistor using ( 5V - 0.7V ) is actually more correct, but it usually doesn't matter, if your base voltage is high enough, since you will always multiply the calculated base current by 2x or 3x anyways. So using 5V or (5V - 0.7V ) is not an issue. Give it a try, and convince yourself that in the saturated switch case, it doesn't matter, especially after you just take the base current calculated x2 or x3 anyways.

It does start to matter when you have much lower base voltages. So if your base voltage was 1.0V, then 1.0V is much different than (1.0V - 0.7)=0.3V. And if you wanted 15ma base current, you can see the difference here in the calculations:

R=E/I = 1.0 / 15ma = 66 ohms, or

R=E/I = 0.3 / 15ma = 20 ohms

So if you ignored the Vbe drop of 0.7V and used the first equation, and put a 66 ohm resistor there, using a 1.0V base voltage, you would have an actual base current of I=E/R = (1.0 - 0.7) / 66 = 4.5ma and the switch would probably not turn on enough to supply the needed collector current. So at lower base voltages you can't take such liberties. At higher base voltages there is usually enough wiggle room that it can be safely ignored.

The other time when you shouldn't ignore the Vbe drop is in amplifier design, where you are not making a simple current switch, and are actually trying to use the transistor in it's forward biased active region.