I think it can appear good on the surface and initially (cheaper tickets wow!) but then for example here in the UK you have newer train operators with special statuses, who have different compensation schemes, who choose to interpret legislation in different ways etc.

eg LNER (an older much larger rail brand) often refuse to carry passengers with a Grand Central (a newer low cost operator) ticket when there is disruption because of internal disagreements between the two companies and different interpretation of the law/contract. This is confusing, annoying and often very expensive for passengers when they board an LNER train (as it’s the next one) and get told they have to buy new tickets

It reminds me of low cost airlines and look at the state of flying these days. Everyone is “low cost” but often without the low cost part. Ie it dragged the bar down so low

No need to look far. In austria the ÖBB and Westbahn are competing. And it turned out quite well.

I believe czech Regiojet also competes in AT.

It looks like it can work if you do everything right, but it can also go very wrong. I think you will always need a central party to coordinate stuff, because letting a bunch of profit-driven railroads arrange this amongst themselves sounds like a recipe for disaster. But that central organisation needs to be run very well, or you still get disaster. Having everything in the hands of a single party has obvious advantages, but also obvious disadvantages. Competition clearly works well enough in some countries, and not so well in others.

I think a thorough analysis of where it works and where it doesn't, and why, would be really interesting.