Here is what I do, The chair is for me. My wife sits on toilet hence killing two birds with one stone. She is unable to brush her own teeth any longer. A dental hygentist informed me that I can use a marginal amount of toothpaste and do not rinse, the benefits of various toothpaste can work if it remains on teeth. Rinse and spit has no true benefit. Just keep the amount small as they will spit if able. You would be surprised how little paste is required as it suds up well. Same for mouth wash. I pour a tiny amount into cap and then soak it up using 2 Q-tips to spread along the gums. When finished I peel down her bottom lip to pour in rest and using my finger to spread along her teeth. Don't care if she swallows the tiny amount if even possible as it spreads well. Shaving....maybe go electric?
Most sinks will not allow a chair to be pulled up in front of it to brush teeth or shave. The cabinet will be in the way. Unless the bathroom has a "vanity" area. Even then the sink is usually not in that area. It might be easier to get a Bedside rolling table and then a basin can be placed on the table and shaving and brushing teeth can be done anywhere. And the Bedside rolling table will come in handy for many other things. Can be used by the couch, another chair, over the bed.
I used a portable kitchen sink or a bed pan, any kind of small plastic tub.
That’s what is done in the hospital when a patient can’t get to the sink.
It might be easier to get a Bedside rolling table and then a basin can be placed on the table and shaving and brushing teeth can be done anywhere. And the Bedside rolling table will come in handy for many other things. Can be used by the couch, another chair, over the bed.
https://www.emso.ca/us/perching-stool-deluxe-20h026.html