Most of the top ranked state schools in DIII have roster caps and tryouts to make the team. Most of the rosters are large but if you have over 100 men that want to be on the team and a 50-70 man roster limit there will be cuts. This is also the case with DIII schools with really large enrollments. I’m sure NYU and it’s 26,000+ undergraduate enrollment has people who are interested in competing but won’t make their team.
Many private liberal arts college do let anyone walk on because they are enrollment driven and are worried about having enough students. When the schools are $60,000+ they can often afford to get an extra bus if size becomes an issue in travel (a full bus of 50 athletes and 4 coaches is $3 million a year in price tag compared to an average charter bus coast at $1500-$2000 per day trip). Some coaches limit because they want smaller teams or they only want to coach a certain caliber of athlete.
No matter what the division most coaches want to be successful. If you are on a team and the coach doesn’t care about success that probably isn’t a team most people want to be on (I chose success and not winning because success can be winning, improvement, development, etc). Whoever said people shouldn’t be DIII coaches if they want to mow down competition hasn’t been around top tier DIII programs. North Central wins against DI teams all the time (and mows down other DIII teams). I’m sure other teams that are balanced like LaCrosse, Mount Union, Rowan, etc. are mowing down their competition in regular season meets.
Most coaches will reply with what it takes to be on their teams. Find a team that focuses on what is important to you.