Are you referring to single shot versus continuous? I'm able to save the drive mode on my R5, just press M-Fn button (it's just in front of the shutter button) and, while looking through the view finder, turn the main dial to view and select the drive mode. Be sure to register the change otherwise it will be lost (it's Set-up, the yellow wrench, 5 on the R5 menus).
I can't register drive mode on my R6III which my guess is pretty similar to the R5II. The R6III's Rate button is now called Color which can be mapped so I assigned Drive mode to it. Does not help much for fast emergency situations so I just have to cull more. While Recall is pretty good it still missing a few features I think. Hopefully Canon expands the Detail Set options for that and a few other things some day.
I use it for BIF situations. I have the cam set up for fast SS and when I press for Recall the shutter speed drops for perched birds. I also have C1-3 mapped for very fast, mid and slow shutter speeds for those situations where I have time to switch. When I had a R5 I could map C1-3 and I used the M-Fn for that. When the R6 bodies you have to manually turn the dial. That is one thing I miss about my R5.
Not sure if Canon can do this for me. Grow me a second thumb so I can press the * for Recall and still get to the QC1 dial for Exposure compensation. There are other options but I've been using the QC1 for exposure compensation for many years. Not sure I can adapt to something else a this stage of my life.