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Synthetic indicators seem to be exclusively used when determining the pH of a substance with an indicator (with the exception of that school experiment where you boil cabbage to demonstrate natural pH indicators) or in acid-base titrations etc. over natural indicators. Why is this the case when naturally occurring indicator can be easily prepared from accessible and sustainable methods, such as boiling cabbage for example? Intuition seems to suggest that perhaps synthetic indicators are much more accurate and reliable, being specifically designed for the task, is this the case? Or are there other reasons why natural indicators are regularly used as a cost-effective, environmentally friendly and easily prepared alternative to synthetic pH indicators?

What are the advantages/disadvantages of using synthetic indicators over natural ones (or vice versa)?

In acid-base titrations, synthetic indicators are exclusively used to find accurate end-point determinations because they always have a highly defined color change at certain pHs. For example, phenolphthalein ($mathrm{p}K_mathrm{a} = 9.7$ at $pu{25 ^{circ}C}$) is colorless in acidic solutions (precisely $0 lt mathrm{pH} lt 8.2$), but it is pink in basic conditions when the pH of the solution goes above 8.2 (precisely $8.2 lt mathrm{pH} lt 12.0$), which makes it ideal for strong acid-strong base titrations (Wikipedia). On the other hand, bromothymol blue ($mathrm{p}K_mathrm{a} = 7.1$ at $pu{25 ^{circ}C}$) is yellow in acidic solutions (precisely $0 lt mathrm{pH} lt 6.0$), but it is blue in basic conditions when the pH of the solution goes above 7.6 ($7.6 lt mathrm{pH}$). However, it shows greenish-blue color in a neutral solution (precisely $6.0 lt mathrm{pH} lt 7.6$; see the picture below).

It also change color from yellow to pink in high acidic conditions such in concentrated hydrochloric acid ($0 gt mathrm{pH}$; see most left test tube in picture below), thus it is ideal to use as universal indicator (Wikipedia). These two example would clearly show you, as you correctly put it, synthetic indicators are much more accurate and reliable since they have being specifically designed for the task.

Sure, natural indicators could be easily prepared from easily attainable row materials, and hence, could be regularly used as a cost-effective and environmentally friendly alternative to synthetic pH indicators, but I regret to say that it can be used only in high school science activities to fascinate science loving kids. Since, those natural sources contain several color changing substances (e.g., flavanones, flavones, flavonols, and anthocyanidins), the color change at the end-points are not sharp, perhaps, due to difficulty of determining $mathrm{p}K_mathrm{a}$ of relevant indicator(s). For example, although boiled red cabbage water can be used as a universal indicator in high school projects and show the end point of an acid-base titration (see following figure), it cannot be used in day-to-day analytical lab to do accurate calculations.

Also, another reason for not using those natural indicator is most of these dyes tends to decompose in higher pH values. The best example is most common natural color indicator, anthocyanidins. The stability of anthocyanidins is dependent on pH. At a low pH (acidic conditions), colored anthocyanidins are present, whereas at a higher pH (basic conditions) the colorless chalcones forms are present (see scheme below) (Wikipedia).

Most importantly, aside from above facts that flavonols including anthocyanidins have been studied extensively in drug development due to their plethora of biological activities (Ref.1). Thus they have been targetted in much more important tasks than usage as pH indicators.

Reference:

1. M. Rahnasto-Rilla, J. Tyni, M. Huovinen, E. Jarho, T. Kulikowicz, S. Ravichandran, V. A. Bohr, L. Ferrucci, M. Lahtela-Kakkonen, R. Moaddel, “Natural polyphenols as sirtuin 6 modulators,” Scientific Reports 2018, 8, 4163 (11 pages) (DOI:10.1038/s41598-018-22388-5).

The main issue with natural indicators is their purity, long term chemical stability and shelf-life. Note that some natural indicators are pretty good for example the famous litmus paper is obtained from lichens. It is a mixture of several pigments like the red cabbage. Sometimes titrations are carried out at warm temperatures, do you think the pigments in cabbage survive heat in acidic and basic solutions?