Most states seem to prohibit private travel unless it is for urgent medical or other strictly necessary reasons. A family visit would not be considered necessary I think unless it is something like a funeral, wedding, or somebody terminally ill in the immediate family.
Also, some hotels strictly limit themselves to business travellers, as you stated. I've been to such hotels and they demand you fill out the form exempting you from hospitality tax. If you file that while that travelling privately, that's tax evasion essentially.
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FLYGVA Just recall back in Nov./Dec. The same rules wrt hotels were in place as today. And several states felt the need to express what constitututes necessary travel over Christmas. If you stayed at a hotel to visit your family back home between Dec 24-26, that was considered necessary travel. Instead, booking a hotel between Dec 30-Jan 2 for a belated family visit was deemed unnecessary and hence prohibited.
Well, if you are witness in a court case, that is required travel, but not a business trip for the witness. In this case, you have to pay city taxes, but of course can provide a letter from court. I am not aware on any problems with this.
A colleague of me needs to visit her 85 year old mother, who is in an old-peoples home and has to deal with paperwork for the old peoples home as well as required work for three tenants with around 28 appartments. She stays once in a month in a hotel for two days and this was okay as well. And as this is a city with city taxes, she pays it as well. It might help, that she does this since three or four years and always stays in the same hotel and on a fixed base (always first Monday to Wednesday in a month). No problems so far, neither in the first lock down nor since November.