At least not entirely.
Back during the Obama era, Republicans pissed and moaned 24/7 about Obama's spendthrift ways.
But a president can't spin one thin dime without Congress first legislating that spending. And absent a successful veto of such spending, he must spend every dime they tell him to (as you may have noticed, trying to condition such spending on "do us a favor," etc., can get him in trouble).
The Republican Party controlled both Houses of Congress from 2011-2019. Every dime Obama spent, the GOP gave him to spend and ordered him to spend.
In point of fact, there was considerable (and justified) saber-rattling by congressional Republicans when some Democrats asserted that, should Congress decline to raise the debt ceiling, Obama could just do it by executive order (the Constitution assigns the power to take on federal "public" debt to Congress alone).
These days, the Democrats control one house of Congress, and they've had enough members in the US Senate to make cloture difficult on spending bills -- if they wanted to -- since before Trump was inaugurated.
Which means that every dime Trump spends, the Democratic Party is giving him to spend and ordering him to spend.
Yes, the president proposes a budget (he's been required to do so, by law, since the 1920s).
No, Congress doesn't have to pay any attention at all to the president's proposal. The next presidential budget proposal passed as offered will be, I suspect, the first one.
And in fact it's been something like a decade since Congress has passed any budget at all, let alone the one proposed by the president. They just keep operating on "continuing resolutions," with periodic "shutdown" theatrics.
I don't doubt that Trump would go on a spending spree that makes past ones look like a cost-conscious visit to Aldi, if he had the power to do so.
But he doesn't have that power. The only extent in which he's to blame for insane spending and ballooning deficits is to the extent that he doesn't veto that spending and borrowing. Which is a little, but not really all that much.