Подходит is more about approaching on foots (it also has other meanings like I wrote). Approaching on wheels=подъезжает. (ходить vs ездить).
Also note that "ко мне" is used, more info here http://new.gramota.ru/spravka/letters/22-spravka/letters/69-ko-dnu (there are listed words to be used with ко).
But still, there is nothing about me in this sentence.
Thanks for the explanation. I just tend to use the estrategy of simplifying as much as it's posible but now I see that приближаться/приблизиться is quite an useful verb.
The second verb exists, the first does not? ))
The other question is interesting. Sources say it would have the meaning "to bring closer" or "to move in". See https://translate.yandex.ru/?lang=ru-en&text=%D0%BF%D1%80%D0%B8%D0%B1%D0%BB%D0%B8%D0%B6%D0%B0%D1%82%D1%8C