The RR portion of the crossing is indeed the responsibility of the railroad but that means Rail, Ties, drainage and concrete or wooden panels. Otherwise the asphalt and everything else of the responsibility of the whomever the road belongs to. Crossing Protection is somewhat the reverse. If its protected by cross bucks only and the road owner wants Gayes & lights installed they go through the FRA and usually try and get a grant to pay for the equipment, which the railroad installs and maintains it going forward.
When I was in charge of rebuilding railroad crossing in Texas as an Asst. Roadmaster my gangs rebuilt almost every crossing between San Antonio and Austin. Interesting learning experience. Later on when I did Operation Lifesaver in Illinois we worked with Govt Affairs and local communities that wanted upgrades.