Sure, but who decides what's related to the original bill? Similar provisions elsewhere (like the "money bill" provision in the Parliament Act 1911 in the UK) work because the person making the decision (in that case, the Speaker of the House of Commons) is strictly non-partisan, but there's no comparable position in the US.
Not having a position or a mechanic to decide such issue is a serious flaw in the system. You gave yourself and example showing that it is possible to have a control mechanism for amendments in a political system.
Yeah, it's definitely an issue.
Not sure if what happens up here is better. Here the budget is simply introduced with all the riders already included. (Typically some of them are things the opposition supports, so that when they vote against the budget--that vote being a confidence vote and thus always whipped--the government can make hay about them voting against something they said they supported. It's also where the government sticks really odious things they think they can slip under the radar, like destroying environmental protections for waterways.)
