I think there's two steps of weirdness. First, many of us see dx/dy not so much as a division of values, but as just some notation for a rate of change strongly linked of f(x). Handling that algebraically seems strange. The explanation of infinitesimals will allow us to sort of begrudgingly accept that such manipulations might be doable, but if we do so we picture dx and dy purely as very small values with a now kind of unclear relationship to f(x).

The integration step then becomes a weird mess. First of all most of us are used to only ever taking integrations with respect to a variable. Except now when we integrate, that 'with respect to' part is somehow already pre-filled by the earlier shenanigans. On top of that, we are somehow now integrating with respect to different variables on both sides which intuitively feels like we just did different things on both sides. And what in the world do our new forms now have to do with f(x)? dx and dy were supposed to relate to a change of f(x), but somehow we just took dy out entirely? At this point we've lost all intuition for what we're doing, and feel like we're just staring at some weird notation hack