IMO there are 'medical' and 'non-medical' sharps.
Non-medical sharps, including broken glassware can be stoutly packaged and thrown with the household waste. Most folk here have the ingenuity to make something sturdy enough not to puncture bin bags or injure waste collectors.
Medical waste carries a small risk of infecting others and needs different management. Appropriate sharps bins are available online (and maybe at pharmacies, I don't know!) for not very much.
The problem is then with collection of these bins. My suppliers collect my bins so 'I'm all right, Jack' but AIUI Councils are supposed to arrange this but it's obviously not happening. All those places now offering Covid jabs must have some way of getting THEIR sharps collected; seems a shame they can't just take the odd extra bin...