Separate rights of way are the only way to ensure on-time service. Bus "bunching" is an unsolved mathematical problem. Traffic is just not predictable, and since buses follow a defined route, they have an even worse time of it than the average driver (who only goes from point A to B, with flexible routing)

This is equally true whether you use rail or bus. A bus system on dedicated busways can be just as fast and reliable as a traditional subway. A rail system running at-grade in traffic will suffer the same low ridership and stigmatization as a regular bus line.

Advantages to buses on dedicated busways: They're slightly cheaper than rail development.

Advantages to rail on dedicated railways: Much greater capacity than a bus system. More "headroom" to upgrade capacity.

Segregated lanes in an at-grade system are a half-measure at best. It gets you nowhere close to the service improvements made possible by a dedicated RoW. A bus system with segregated lanes is just a slightly faster bus system. A bus system running on dedicated busways is rapid transit.