It is also worth remembering that a mechanical license is limited.
It allows you to perform a work yourself (privately) to create a new audio recording of that work[1]. What is known as a "cover". But other than sell the new cover audio as CD, mp3, etc, or have it get played on the radio or audio streaming services, there is not much more you can do.
It does not allow your cover band to perform the work publicly. (Normally this gets covered by the venue, either via ASCPA/BMI licensing, or via individual negotiation with the rights-holder, but in some cases, like temporary venue fully orgnaized by the band, the band may need to acquire such rights.) Remember that any form of live-streaming is also public performance.
It does not allow your cover band to create a music video, or to use your cover recording as background music in a video you create. (For either of these you need a "synchronization license" that the work holder can simply refuse, and there is no simplified ASCAP-like mechanism for handling these negotiations)
It may also surprise many to learn that recording artists are not technically entitled to royalties for use of their songs over the radio. Indeed, recording artists of any kind[2] only get royalties for digital music transmission when certain exemptions don't apply, and get no protection at all for non-digital transmission.[3]
The whole music copyright system is basically a giant mess, and does not work how a reasonable person would expect it to work.
Footnotes:
[1] Or to duplicate an existing audio recording if you also have permission of owner of the copyright for that particular audio recording.
[2] unless they happen to be the musical work copyright holder as well, which is not uncommon, but hardly universal.
[3] Realistically they were only granted this digital public performance right, because a digital transmission of a digitally recorded phonorecord is hard to distinguish from copying said phonorecord.