No the Dems are the ones who traditionally want large spending. Going back into the 20s. Republicans have undoubtedly jumped on the spending band wagon. The point isn't who started it or who has been the big spenders I. The past. It's who's doing it now that matters. And both sides are as guilty of it. You are right that the Republicans seem to talk a big game, but when the rubber meets the road they spend like the Democrats. No difference. When we see actual cuts in spending then I will be impressed. Then I will take talk about the concern for the deficit seriously.
I don't think that you can point to one party or the other as "traditionally wanting large spending". While you can point to Woodrow Wilson (a Democrat) as the first president to run a serious deficit, that was in response to WWI. The entirety of his deficit was accrued from 1917-1919 - the other years he broke even or ran a surplus, including his final two years in office. Harding and Coolidge (both Republicans) ran surpluses, but Hoover (also a Republican) ran a deficit as he dealt with the Great Depression.
Roosevelt is probably the most famous Democratic deficit spender, but much like Wilson, the bulk of his deficit spending was related to WWII. He did run deficits from 1934-1941 (aside from 1938, when he broke even) as he dealt with the Great Depression and its aftermath, as well as supporting our European allies at the beginning of the war, but they were comparable to the deficits run by Hoover (again, a Republican) at the end of his administration. It wasn't until 1942 that the annual deficit ballooned from 5 billion to over 20 billion, then about 50 billion for the remainder of the war.
Truman (also a Democrat) ran surpluses for about half his presidency for a very modest overall increase in the deficit. Eisenhower (Republican) was about the same. Kennedy (Democrat) ran a deficit every year, but not a large one. Johnson (Democrat) ran similar or lower deficits to Kennedy aside from 1968 (Tet Offensive) and even managed a surplus in 1969.
Nixon (Republican) ran a larger deficit than Johnson pretty much every year as the US ramped up involvement in Vietnam. Ford (Republican) ran huge (for the time) deficits of ~50 billion - similar to Roosevelt's spending during WWII. Carter (Democrat) followed the same trend, with a slight increase, but then Reagan (Republican) turned it up again, more than doubling - nearly tripling, even - Carter's highest annual spending most years. Bush Sr. (Republican) ran even higher deficits than Reagan, but Clinton (Democrat) was the first President since Coolidge to run an overall budget surplus over the length of his time in office.
Bush Jr. (Republican) promptly turned that around and returned spending to the same level as his father, and even higher, with the invasions of Afghanistan and Iraq. Obama (Democrat) blew up the budget even more with his response to the Great Recession, but was starting to bring it back under control towards the end of his second term. Trump (Republican) has returned spending to levels near what Obama's were at the beginning of his first term.
Overall, presidents of both parties have grown the deficit. However, from this data, what I see is that (aside from Carter) every Democratic president who has significantly increased the deficit compared to his predecessor has done so in response to a new war or a recession/depression - or both. On the Republican side, Reagan, Bush Sr., and Trump have all presided over significant deficit increases without impetus from war or recession (I don't count the '91 Iraq invasion as a war, since that only lasted a couple days).
Which President Rang Up the Highest Budget Deficit?