It's good to understand media as either defined (synthetic) vs. undefined (complex), and minimal vs. rich. Some consider 'rich' to be synonymous with 'complex,' but I believe this is incorrect.

A defined, or synthetic, medium is one in which all the components and concentrations are known. In complex media, you don't know what the exact concentration of all the components are. The chief example is LB. While you may make LB the same way every time, the yeast extract and tryptone have likely slightly different compositions of peptides in every batch you buy from the supplier.

A minimal medium is one in which only few and necessary nutrients are supplied, such as a carbon source, a nitrogen source, salts and trace metals dissolved in water with a buffer. Minimal media is thus a synthetic medium, because you define all the concentrations. 

A rich medium is most easily defined as a medium that supplies more nutrients than a minimal medium! You can have a synthetic rich medium, for example, by supplementing a minimal medium with additional amino acids and nucleobases. Or, you can have rich complex media like LB, which has lots of nutrients though you don't know exactly what they are. 

To answer your question, I think the main reason people use complex media is that it's easy. Cells grow fast because the media is rich, the ingredients are cheap, and it's easy to make. People grow E. coli, which also grows fine in minimal media, in LB all the time. 

You should be able to make a synthetic medium for your favorite fastidious microbes, but it will likely be a huge pain, so you might as well use a complex medium unless you need to control a certain nutrient input for some reason. 

If you want further reading on developing media, especially as it applies to bacteria, you should look into some of the old papers by Neidhardt. 

It's good to understand media as either defined (synthetic) vs. undefined (complex), and minimal vs. rich. Some consider 'rich' to be synonymous with 'complex,' but I believe this is incorrect.

A defined, or synthetic, medium is one in which all the components and concentrations are known. In complex media, you don't know what the exact concentration of all the components are. The chief example is LB. While you may make LB the same way every time, the yeast extract and tryptone have likely slightly different compositions of peptides in every batch you buy from the supplier.

A minimal medium is one in which only few and necessary nutrients are supplied, such as a carbon source, a nitrogen source, salts and trace metals dissolved in water with a buffer. Minimal media is thus a synthetic medium, because you define all the concentrations. 

A rich medium is most easily defined as a medium that supplies more nutrients than a minimal medium! You can have a synthetic rich medium, for example, by supplementing a minimal medium with additional amino acids and nucleobases. Or, you can have rich complex media like LB, which has lots of nutrients though you don't know exactly what they are. 

To answer your question, I think the main reason people use complex media is that it's easy. Cells grow fast because the media is rich, the ingredients are cheap, and it's easy to make. People grow E. coli, which also grows fine in minimal media, in LB all the time. 

You should be able to make a synthetic medium for your favorite fastidious microbes, but it will likely be a huge pain, so you might as well use a complex medium unless you need to control a certain nutrient input for some reason. 

If you want further reading on developing media, especially as it applies to bacteria, you should look into some of the old papers by Neidhardt. 

In short I think the statement in the question is false.  The optical density that a culture attains in media (especially in a shake flask environment) is usually governed by the accumulation of waste products as opposed to nutrient depletion. A 'fastidious' microbe could actually find a particularly rich complex media environment stressful.  In my experience, a well designed synthetic medium (which need not be minimal) will outperform a complex medium.  

Probably the main reason for this is that in a synthetic medium inevitably has less micro nutrients and the culture must therefore build more things from scratch.  Many of the synthesis pathways involved in this get their raw materials from the TCA cycle and therefore compete with acetogenesis.  A well deigned synthetic medium will also facilitate the reassimilation  of acetate and speed up any rate-limiting steps that could be detrimental to your application.  I have attached some papers that are good for getting to grips with this.  They are specifically about E.coli, but obviously much these pathways are pretty universal.

Having said all this, I agree with John Sauls;  developing a great synthetic media is resource consuming and if you are just trying to grow up some bugs for cloning work etc. then complex media is probably a better choice.

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The statement is perfectly true. Complex media provides nutritional requirements for fastidious microbes  because they have complex cultural and nutritional requirements. This works because in a complex media, the chemical constituents are not clearly defined 

It is important to as John said to make the distinction between complex and rich.  A synthetic media can be very rich and provide all the important micro-nutrients with the added benefit that the concentration of said nutrients is known.  It is entirely possible to develop an synthetic media that provides everything the microbe needs in correct abundance making it effectively just as rich or even richer than a complex counterpart.

People use complex components such as yeast extract because they correctly assume that a microbe extract will be a rough and ready source of micro-nutrients and trace elements in roughly the correct abundance.  If you are just trying to grow something up, or reactivate a freeze dried culture, then complex media is a good choice for this reason.  But just be aware that there is such thing as too rich and some microbes find high growth rates stressful.  Additionally; complex media, by virtue of providing many of the building blocks for log phase is more likely to result in aerobic metabolism over-spill and result in the accumulation of acetate, lactate etc.