This may fall out of the scope of the sub. Considering the parties switched plat forms in the 60s, looking from 1970-Today gives a more accurate picture. This is more recent than this sub likes to go, but I've analyzed the historic data a decent amount.

Side note: Trying to compare modern day republicans to the Lincoln era is comparing apples and oranges; they are both fruit/political parties, but they are not the same party (despite the same name).

Going back to your original point:
Saying that a president is responsible for the debt is misleading. The house submits the budget, which gets reviewed by the senate, and signed by the president after everyone else agreed. The president is 1 part of the puzzle. It is rare that 1 party controls all 3 in a given year, so it is rare that a party is ever single-handedly responsible for the budget. If you want to look at it that way, the debt accumulated almost exactly 50/50 based on the party of the sitting president from 1970 through 2019 (the most recent year when I actually did this analysis). We can take this a step further and state under the last 6 republican presidents, none dropped the deficit. However, 4/5 democrats did, and 2/5 left with a surplus! Again, misleading, but it is line with what you were looking at.

A better way is to look at how states are run or what happens when the party controls all 3 branches.

If we look at the last time the dems held all 3, 2009-2011, they raised the deficit. However, they raised and then dropped it. They specifically rose it to counteract the market crash of 08. Before that was 1993-95, which they decreased the deficit by roughly a third. Republicans on the other hand held all 3 branches in 2017-2019 and raised the deficit by 50%. This was during a booming economy, which is when the deficit and debt should be reduced. Before that, there was a period where they had control somewhere in the 2001-2007 time frame. The data got muddy so I could not confirm what years they had complete vs partial control. It seemed like they decreased the deficit while they had control, but I cannot say for certain either way. The score is dems 1/2 for decreasing the deficit, repubs are maybe 1/2 for decreasing. Going back 50 years leaves republicans the same, dems at 1/3.

Republicans view of "less government" seems disconnected from their more spending policies and budgets. Anyone arguing they are currently a party of low budget/debt has a tough argument to make.

Now an argument can be made they are the party of less government because they want to defund public institutions and subsidize private business. They tried that this year with the post office, they've tried this in recent years with pushing for charter schools. They have also tried lowering restrictions on things such as labor laws, environmental regulations, and even tried making it so employers are not responsible for keeping their work places safe from covid.

TLDR: The data does not support republicans are a party of small government when discussing dollars. An argument can be made they are still for small government.

Edit: Cleaned up the diction and syntax. Added some more info.