I send proofs on everything, largely to try to get them to look one last time before printing, and review any corrections I had to make to get their job print ready, and confirm specs / layout. Reality is - my printed color off a digital press is unlikely to exactly match what they are looking at on their phone, and they aren’t often the designer or if they are, they aren’t often “a” designer so unless they have made specific callouts and branding expectations then the color reproduction defers to our expertise.
I find putting in a serious disclaimer a bit customer unfriendly - if your customers are businesses then they should know what they are ordering - a bit like I don’t give someone a PO until I know what I’m asking for and willing to pay for it unless the vendor makes an error. Assuming we are talking about digital run lengths, not magazines, etc…huge runs that would be very expensive and tricky to fix. Similarly I don’t think it’s right if the printer makes an error (typographical or quality / spec error) and argues with their customer “bUt yOU aPpRoVEd iT liKE tHIs”. If the printer made the error, then yeah you generally should still make it right in most scenarios.
If your customer is the general public, then yeah you can’t assume they know what an approval process is and you probably have to be more firm with them.
In either case, I do also feel like many customers are not taking the time to thoroughly review proofs and it’s probably a better practice to point out any issues you notice to them than to hide obvious flaws just to be pedantic.