Some bacteria replicate every 20 minutes. That's 72 opportunities a day for them to catch onto at least the beginnings of a method to bypass an antibiotic. And mutations are to increasing environmental survivability as brute force cracking is to opening a file with 2056-bit XYZ+ encryption. It'll work eventually, but 99.99999% of the time (literally) you and your entire family tree are long dead before anything significant happens.
You are underestimating the mutation rate of bacteria, as vertical inheritance is not the only mechanism for mutations. Look up Horizontal (or Lateral) Gene Transfer. Living cells can acquire genes from other cells. In bacteria, it was measured at one successful transfer per generation (in E. Coli). Genes can be acquired from environment, through plasmids, injected by viruses, recombined with other bacteria (a.k.a. bacterial sex), etc. Transfer doesn't have to be from the same species (whatever definition of species you use). Transfers can be cross-species, cross-phyla or even cross-domain. A useful trait like resistance will spread very very quickly.
The problem with antibiotics is that they a very strong selection pressure on bacteria: they kill them. And a stressed cell will have enhanced mutagenesis (look up stress-induced mutagenesis). No living being likes to be killesd, so we all have mechanisms to adapt to that situation. When you have anitbiotics, not only the bad bacteria can develop resistance. Also the harmless bacteria can develop resistance, and then transfer it to bad ones.
An alternative is not to kill bacteria, but to trick bad bacteria into expressing their pathogenicity (a very energetically demanding process) when their numbers are very small, so the other harmless bacteria can outnumber them to irrelevance. A way to do this is called Quorum Sensing Spoofing (hijacking of inter-bacterial communication), which is in very early research stages.