The usual pictures of и / п / т / ш ambiguity that you see are exaggerated in that they show forms that are nominally “standard” but basically impossible to reproduce without a fountain (or, even better, dip) pen (think round hand or, as 'cyberax mentions, Spencerian script), yet use a constant stroke width that such an implement wouldn’t produce. For the latter two, people who actually write m and not т will often resolve the ambiguity with ш with an over- resp. underbar (the same ones that Serbian uses even in print[1]). It’s also pretty normal to exaggerate letter joins when they come out looking too similar to parts of other letters, etc. Overall, modern Russian cursive is about as legible as the modern French one, and I don’t think people complain much about the latter.
I also find the hand-wringing about English accents somewhat surprising. Yes, different accents exist, and yes, English has a much wider variation than (urban) Russian (there are things in the countryside that urban dwellers haven’t heard for a century), but phonemic orthographies are a thing, and though children in e.g. Moscow may perpetually struggle with orthographic distinctions that no longer correspond to anything in their accent, the idea of a spelling competition remains about as laughable as that of a shoelace-tying one. Nobody makes you represent the many mergers of English with a single letter in your new orthography (though it would be funny).
[1] https://commons.wikimedia.org/wiki/File:Cyrillic_alternates...., rightmost column