Many, if not all, stars will, in their birth phase, have an accretion disk, or circumstellar disk, around them, formed of the material from which the star forms. This disk dissipates in a few million years, both due to the material accreting onto the star, due to material being blown away by radiation pressure, possibly assisted by dust grains, and due to depletion of the material to rocks and planets. Friction in the disk causes it to heat up and be observable in the infrared. After this phase, the star usually won't have an accretion disk. That is, until they die.
As you suggest, a star that later on goes supernova can accrete matter from its "reverse shock", i.e. matter that is falling back due to the shockwave hitting the surrounding interstellar medium. More often, though (I think), will the accretion disk be made of material from a companion star: If one star of a binary system turns into a black hole, and then the other star starts to evolve into a red giant, threspassing the Roche limit, it will transfer matter to the black hole.
As you say, the black hole's mass isn't stronger than the original star's mass, and thus the gravity at a given distance is also not stronger (in fact it's smaller, since it has blown out most of its mass in the explosion). However, due to its much smaller size, the accreting matter can come much closer before being sucked in. Coming much closer means being in a much steeper gravitational potential and reaching much higher speeds, i.e. being heated to much higher temperatures. Hence, black hole accretion disks can be detected in X-rays
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