The answer to your question is “no.”
Sure, the standard infectious disease way to go about things is to begin with culture and stain. Stain, meaning to take the clinical specimen, add dyes, and look under a microscope. Culture meaning that we can take the clinical specimen and using fancy algorithms and culture media, determine what grows and how it looks to decide what bacteria and in there. Nowadays, of course, you just draw blood into a few liquid culture bottles containing who knows what reagents and a few days later you stick it into a fancy computer which identifies which bugs are in there.
Culture and stain is good for diagnosis of a lot of the big illnesses – namely urinary tract infections, strep throat, pneumonia, tuberculosis (no culture, just stain), and others. There are many reasons why we can’t use it – the causative agent is too small for staining or is invisible to stain (like viruses, rickettsial diseases, and Chlamidya for example). Or it doesn’t easily culture (like viruses, Treponema pallidum, TB). In theory, culture can detect if only one bacterium is present. In practice, since everything is covered with bacteria, it doesn’t work like that even for hardy bacteria. This is particularly true of a disease like typhoid which may wax and wane and have lots of variability in the clinical samples. Certain clinical samples are more difficult to deal with than others (for instance stool). And while staining is fast, culture takes at least 3 days. So other strategies (immunoassays and PCR specifically) have been designed. Immunoassays are routinely used for rapid diagnosis – you mentioned two (the monospot test for mono and the strep throat swab). In-office rapid strep tests take 15 minutes. PCR is the most sensitive assay out there, it can be quite fast, but it is usually something processed by bigger diagnostic labs. So this slows things down.
If you properly design a PCR strategy, you can detect one piece of DNA in your input sample. The key, obviously, is the “proper design” part – obviously even if you have a little wiggle in your design, if you have a few billion bacteria to begin with, a 10,000 fold weed out is still incredibly noisy. Luckily, good PCR strategies aren’t particularly difficult to construct, especially with the large-scale sequencing of many bacterial genomes. PCR works by amplifying DNA between two designed short pieces of DNA called primers. If the primer sequence exactly matches your bacteria of interest and nowhere else in nature (which is not uncommon given evolutionary divergence), then your PCR will amplify only your region and nowhere else in nature. Naturally, if you don’t have your bacteria in your input, you won’t get PCR product. If you do, then you will. Voila, diagnostic test. Add in FDA approval, lots of science about sensitivities, specificities, failure rates, etc, and you are ready to take it to market.