逃げる literally means "run away", which is natural to be intransitive even if you speak English. It also takes an argument with ～から, that corresponds to "from".

逃れる has a few different shades of meaning.

1. To keep away from a certain point where a (clear and present) danger exists. The "point" could mean a bomb, a burning building, or a man wielding his knife etc. In this sense, we use [point]-から逃れる.
2. To go out of (escape) a hostile area. "Area" means that it is a certain expanse of space, which could be an unstable country, a concrete jungle, or the state of poverty. In this case, we usually use [area]-を逃れる.
3. To avoid possible bad consequence in the future, which is not realized at this moment, but easily imaginable as a menace. Like arrest, public blame, or bankruptcy. It is also used in [consequence]-を逃れる.

～を逃げる sounds just wrong to me, although I can suppose that some people might use such a construction by analogy with 逃れる.

So your last sample phrase 役員を逃げた is pretty much ungrammatical, while 役員を逃れた is understandable in the way you evade being appointed executive.

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**Edit**:  
After some discussion in the [chatroom][1], the transitive ～を逃げる might be a kind of older wording. It is very rare in [BCCWJ][2] (1976-) and have some appearances in [青空文庫][3] (out-of-copyright books). Modern speakers would not recognize this usage. We more naturally reword them in the present-day language.

> 役員を逃げる → 役員を回避する (?)  
> 税そのものを逃げる → 税そのものを逃れる  
> 網の目を逃げている → 網の目をかわしている  
> 胸苦しさを逃げるために → 胸苦しさから逃げるために

Note that there are another "locational" を, which means "through" or "via", and does not make a verb transitive.

> 地下道を逃げるのよ！早く！  
> _Run (away) through the basement (passage)! Hurry!_


  [1]: https://chat.stackexchange.com/transcript/511/2020/9/19
  [2]: https://pj.ninjal.ac.jp/corpus_center/bccwj/
  [3]: https://www.aozora.gr.jp/