Imitation of an older convention
It's clear that the designers of more recent currency symbols have their own rationale for including the slashes or 'strikeouts' in the symbol. It's also clear that these elements naturally evolved in older currency symbols through the use of abbreviation and shorthand.
It's more than likely that modern currency symbols are using this older convention, that originally evolved naturally, that slashes through a symbol indicate currency. This makes it much easier to distinguish from regular characters. As you said in your question, the practice has arguably become a pseudo-standard.
The Euro is a relatively new currency and the symbol was designed and presented in 1996. The explanation of the design given by the European Commission (emphasis mine):
Inspiration for the € symbol itself came from the Greek epsilon (Є) – a reference to the cradle of European civilisation – and the first letter of the word Europe, crossed by two parallel lines to ‘certify’ the stability of the euro.
A number of explanations for the crossbars are given in Udaya Kumar's design proposal for the Indian rupee sign:
The use of Shiro Rekha (the horizontal top line) in Devanagari script is unique to India. Devanagari script is the only script where letters hang from the top line and does not sit on a baseline. The symbol preserves this unique and essential feature of our Indian script which is not seen in any other scripts in the world.
The two horizontal lines with an equal negative white space (imaginary space) between them create a foreground and background effect of three strips (tricolor). The strips subtly represent the tricolor of our Indian national flag flying at the top.
The horizontal lines also denote the arithmetic sign ‘equal to’. [...] The arithmetic sign denotes that relationship of comparison of currency values. The equality sign also signifies a balanced economy, our economy should be secured and stable forever.
The origins of the Dollar symbol are somewhat more debated than more recent symbols. It originated when English Americans were trading with Spanish Americans in the 1770s.
The most credible theory is that it is derives from the abbreviation 'ps' for the Peso. Another theory is that the symbol originated as an '8' with a slash through it—denoting Pieces of eight or the Spanish dollar.
Theories for the origins of the dollar using two vertical lines include the idea that it originated as the abbreviation 'US' or as a representation of the Spanish coat of arms, which showed the Pillars of Hercules with a banner curling between them.
In The American accomptant, published in 1979 (a great find by @Yorik!), you can clearly see the notation used for 'Federal Money'. 1 Cent is expressed as
//. 1 Dime overlays an
S over the
// and 1 Dollar overlays a double stroked
//. This calls in to question the other theories of the Dollar symbols origin. The document doesn't however give any explanation for the notation (As far as I can tell—I haven't read all 320 pages!)
The Pound symbol is more obviously derived from a cursive majuscule 'L', which represented 'libra', the basic unit of weight in the Roman Empire.
The slash[es] through the Pound come from scribal abbreviations which were in common use in the Roman Empire and using shorthand became common around this time.