Republicans seem to have been pushed further right, moderates from either party are a dying breed.
I'm not optimistic about seeing an increase in moderate behavior in the republican party given the increased divisive state of politics. The guy was not perfect, not by a long shot but compared to what you expect from politicians I think he did okay when drawing lines on some partisan issues. He definitely failed on some fronts but it's refreshing to see some of this.
The nation was more concerned about laughing at him eating all that food in NYC than policy initiatives I guess.
From ballotpedia:
Kasich received both support and criticism from fellow Republicans for his more moderate political philosophy. He supported Common Core education standards, a path to citizenship for illegal immigrants, and Medicaid expansions in Ohio.[13]
Kasich declined to endorse Republican presidential nominee Donald Trump, and on October 8, 2016, after The Washington Post released a 2005 video of Trump making comments about women that the Post described as "extremely lewd," Kasich announced he would not vote for Donald Trump in the 2016 general election.[9][10]
Medicaid expansion On February 4, 2013, Kasich announced that he would participate in the federally-funded Medicaid expansion program introduced by the Affordable Care Act. Kasich's announcement was criticized by members of groups opposed to the ACA, many of whom had supported Kasich in the 2010 election. Americans for Prosperity policy manager Nicole Kaeding argued that "Medicaid is a broken, costly system that needs meaningful reform; expanding the system to include another 365,000 individuals is exactly the wrong policy for Ohio families."[46] Kasich received permission from the state controlling board to implement the program on October 21, 2013. Kasich's decision was praised by the White House, with Obama aide Chris Jennings saying, "this is great news for the 270,000 previously uninsured residents, the states health care providers, and [the] Ohio economy, [which] will benefit from this expansion."[47] https://www.cleveland.com/opinion/index.ssf/2018/08/gov_john_kasichs_lasting_legac.html
Veto of Republican-backed residency proposal/voting rights In April 2015, Kasich used his line-item veto power to strike a proposal that would have required out-of-state residents to obtain an Ohio driver's license and vehicle registration within 30 days of registering to vote. Kasich also vetoed a provision that would have taken away driving privileges from new residents who did not meet the 30-day deadline for obtaining in-state driving documents. Ohio Democrats opposed the provision, claiming it was a move by Republicans in the legislature to suppress a segment of the population that often voted Democratic, and commended Kasich's veto.