Experienced business travelers want to use their personal credit cards so they get the points, and scaled businesses overwhelmingly disallow this, because they want that value for themselves.

I'm cool with my employer keeping any spending rewards that come from their spending.

I get that leaves money on the table, but it also eliminates the possibility of employer not paying me back for whatever reason.

My last company had the best of both worlds, they paid the card directly provided you submitted your expenses reports in time and allowed you to use the points. The CEO overrode the bean counters on this.