In theory, the House and the Senate merely tweak the White House budget, whose main use in essence is to sketch out the latter's priorities. And frankly, that's more or less what occurs when the House and Senate are on the same wavelength as the White House.
In practice, the three aren't on the same wavelength more often than not. This results in a lot of haggling between the three to get things passed, and at times rewriting of the budget in bulk by the House and/or Senate as you note in your question. As you'll see from the pdf chart in the previous link, in recent memory:
- Democrats were in control of the House and Senate for the better part of the 1930s to 1990s.
- Reagan and GHW Bush never had both on their side.
- Clinton, GW Bush, and Obama only had both on their side at the beginning of their respective tenures.
- Trump (currently) has both on his side - but with an extra twist discussed below.
Put another way, in recent memory it was typical for one or both of the House and Senate to not have the same political leaning as the White House, resulting in intense haggling between the three as mentioned above. And since it is Congress that holds the budget purse at the end of the day, the White House is often on the receiving end of a budget rewritten by Congress.
At this point, the rise of the Christian Right in the 1970s and the more recent rise of the Alt-Right is worth a mention in my opinion. Neither group is particularly compromising, to put things mildly. That makes reaching a consensus harder and even more prone to legislative pissing contests, where this or that pet item gets funded or defunded so everyone can come back to voters with a pound of flesh. (This is not to say that traditional Republicans and Democrats are easy to deal with; rather, it is that dealing with the two hardline groups is even harder.) Notice, in passing, that this coincides with the House and Senate turning red, which I'd argue is indicative of the notion that the bolder they are the better they fare.
At any rate, Trump's case is a bit special in my view because, in contrast with Reagan or the two Bushs, who had the backing of traditional Republicans with the Christian Right latched on as a bonus, Trump primarily has the backing of the Alt-Right and the Christian Right, with the traditional Republicans (somewhat disgruntledly) latched on. What more, he has been divisive enough since the beginning of his presidency that he's facing resistance within the Republican party. As such, he ends up in the odd position of needing to reach compromises with the House and Senate even though the latter are both (nominally) in his team.