In the interest of fairness and nervous parents whose kids use that bus service, that's Bluestar, which is a public routed bus service, with additional school and uni etc routed services, not a hire and reward coach trip company doing one off coach journeys.
There is a difference, and you expect a higher standard incidence rate with a public service company running city center and outlying bus routes, but in terms of licensing, supervision and traffic commissioners, it's similar enough.
You point at them as being a safety concern, so it seems only fair to show their operations record of actual serious incidents. (If they where cowboy operators, I'd be one of the first to point fingers.)
In the last 18 years, and thousands of uneventful journeys, their serious incident roster is:
2 double deckers have hit low bridges on unfamiliar temporary routes while on rail replacement service.
A car drove into one of their buses, sadly ending the life of the car driver.
2 buses (one not in service at the time being tested by mechanic) caught fire.
Fast forward to 2020's:
4 years ago one bus rear ended another, when the first took evasive action.
June this year the most serious in terms of alleged preventable safety issues:
a bus on the 607 service for Barton Peveril College allegedly suffered a jammed accelerator and simultaneous brake failure, causing it's driver to crash off the road and end upright in the river beside the road. Allegedly after managing to miss all oncoming traffic.
Recently the no 18 and a DPD van were involved in a non serious (defined by no injuries) collision - fault apportioning is I believe still under review.
A (Bluestar) unilink bus was hit by an E-scooter leaving the rider injured. Allegedly the Escooter rider failed to give way at a junction and crashed into the side of the bus, seriously injuring her arm. The police are not disputing this version of events.
If these are the recent incidents, they don't come under serious incidents in terms of a bus company at being at fault, they're standard traffic accidents and police traffic investigations. If they don't like any of what they find it goes up to serious incident and up a level. Have that a few times and never mind insurance, it becomes an operations licensing and traffic commissioner issue.
Accelerators jamming and subsequent brake failure are a very serious incident and require higher investigation into how that can happen, and ensuring it can't ever again.
Like most, they've never had a bus roll over.
They operate 29 general routes, and 40 contracted school routes, including 31 for the College and while not a coach company, that's 18 years and many thousands of journeys and miles worth of serious incident history.
I'd say your statistical chances of traveling safely on one of their buses, on one of their routes, are pretty good.