This is a very naive take - the very direct translation of what you're saying doesn't happen does in happen in analysis all the time: there are many inequalities which can be "stacked" to prove a bound on something but their factors are too large so you cannot just stack them if you need a fixed bound for your proof to go through. Unsurprisingly this is exactly how actual runtime analysis also works (it's unsurprising because they're both literally math).
