MK we have been in the flooring business for over 30 years. I am assuming you replaced the vinyl with Ceramic tile not vct tile. If so, you don't necessaritly have to use quarter round. You can use vinyl cove base, you can sit your base right on top of the ceramic tile, or use quarter round. However, you need to caulk in between the tile and whatever you choose to sit on top of it. You can get colored caulk to match the floor or wall or base. If the quarter round you use matches the rest of the house it will more cohesive but that is only if it compliments the colors in your tile. If it doesn't then match the quarter round to colors that are in that room.

Worthy your comments are not worthy. That is not shoe molding it looks like window stop molding from HD. Looking at your door to floor gap I will assume they cut the door frame moldings and installed the base on top of the floor install? They probably were too lazy to scribe and plane their baseboards to your floor. They saw the dips and large gaps and grabbed some 1/4 round as it is easy to bend and conform to your floor. Hire a real carpenter and have new door casing and baseboards installed. In the end you won't send much more and you won't wind up with all those small returns of so called shoe molding. ANYTIME you use 1/4 round you are covering up a mess and making a larger mess. Shoe molding has a different profile and is not needed unless you are in the NE. If you do use it I agree to forget about all the returns and just cut back at an angle and be done.

Quarter rounds are fine; but, do the white match? In all my years of working with tiles, I've matched different brands of tiles and edging very few times. Consider a Schluter or similar for the edges.

@Jo Reno the job of baseboard, moulding, whatever aesthetically is to provide a transition. Typical used in a more traditional home. In a modern home it is possible to go "naked". But the meeting of floor and wall had better be damn good quality workmanship.  Functionally it can help protect the wall from the dings and scratches associated with cleaning and maybe some aspects of living.  In a bathroom, there is no functional need between a tile floor and tile wall or half wall. In my sort of traditional bathroom with stone floor and some tile some painted walls, and having a desire to make the room feel longer, I used the edge of countertop material cut as a baseboard even into the shower. The top edge did get caulking (which I agree is standard). And by the way, I have never seen an absolute straight wall in any of my homes. That's why these craftsmen should be appreciated.  The miracles many of them perform...