1) Hence the word trying. If he fails but guts the department to the point it can't run, that is in practice doing the same thing. The problem with the whole claim it's the DoE that's the problem and it should be sent back to the states is that the majority of education "stuff" is already controlled at the state level. It's part of the reason there is a gulf in educational achievement between various states. Republicans regularly attack the DoE and say the states can do it better yet Republican states dominate the bottom of the table for educational attainment whilst Democrat states dominate the top. If Republicans had the answers, why are thy not enacting them at state level? With the whole devolution to the states argument, that would make more sense if it wasn't for the fact Republicans are fully on board with riding roughshod over the states when it suits them. A recent example is Trump overriding California's water policies and emptying their reservoirs supposedly to fight fires. It did nothing to help fight the fires, nearly flooded land and now the water supply needed for irrigation is gone. Republicans applauded that direct federal intervention in a state matter. Republicans also regularly attack institutions of higher learning and denigrate higher education as "woke", including spreading outright lies about what is taught at uni.
2) Regardless of whether the DoE did a good job and was fit for purpose, there is a legal way to do things and an illegal way, Trump is doing the latter. Congress has the power to allocate funding, not the executive. The original intent of the founders was that Congress would hold the most power and the executives role was largely to enact and coordinate what Congress passed. Nixon famously tried to simply not fund projects Congress has assigned funding for and the supreme court ruled the president didn't have the authority to use the executive to simply not administer the funding appropriated by Congress. DOGE also falls foul of this, it's not their place legally to simply cut funding in places, that's Congress's job.
3) I did actually criticise Obama for that. The difference was, Obama and later Biden have dealt with some of the most obstructionist Congress in history. The US government was brought to the brink of shutdown more times under Republican control under those 2 administrations than every other Congress combined going back to ww2. The last Congress was officially the least productive Congress in the history of the USA, passing less legislation and getting less done than even the Congress of the earliest days of the republic.