There are two issues here
address is just a convenience alias for BibTeX compatibility. Internally the field is called location (address is automatically converted to location by Biber), so if you want to delete the field on the biblatex side, you need to delete location and not address;
location (address) is a list and not a normal field, so you need to use \clearlist instead of \clearfield (biblatex has three field types: normal fields, lists and name lists, each of those types has its own \clear... command: \clearfield, \clearlist, \clearname, you can look up the type of a field in the biblatex documentation, §2.2.2 Database Guide > Entry Fields).
So
\AtEveryBibitem{\clearlist{location}}
\AtEveryCitekey{\clearlist{location}}
would work.
In this case, however, I think that the \clearfield/\clearlist approach is inferior to other approaches to delete fields.
Either use a Biber sourcemap (which comes in before the address->location remapping and thus needs to delete address and location)
\documentclass[american]{article}
\usepackage[T1]{fontenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[notes, backend=biber, giveninits=true]{biblatex-chicago}
\DeclareSourcemap{
\maps{
\map{
\step[fieldset=month, null]
\step[fieldset=address, null]
\step[fieldset=location, null]
\step[fieldset=doi, null]
\step[fieldset=url, null]
\step[fieldset=isbn, null]
\step[fieldset=number, null]
}
}
}
\begin{filecontents}{\jobname.bib}
@book{Gardner2018a,
address = {Oxford},
author = {Gardner, John},
publisher = {Oxford University Press},
title = {From Personal Life to Private Law},
doi = {10.1093/oso/9780198818755.001.0001},
year = {2018},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
Lorem \autocite{sigfridsson,Gardner2018a}
\printbibliography
\end{document}

Or delete the fields when they are read from the .bbl file with \DeclareFieldInputHandler.
\documentclass[american]{article}
\usepackage[T1]{fontenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[notes, backend=biber, giveninits=true]{biblatex-chicago}
\DeclareFieldInputHandler{month}{%
\def\NewValue{}}
\DeclareFieldInputHandler{doi}{%
\def\NewValue{}}
\DeclareFieldInputHandler{url}{%
\def\NewValue{}}
\DeclareFieldInputHandler{isbn}{%
\def\NewValue{}}
\DeclareFieldInputHandler{number}{%
\def\NewValue{}}
\DeclareListInputHandler{location}{%
\def\NewValue{}}
\begin{filecontents}{\jobname.bib}
@book{Gardner2018a,
address = {Oxford},
author = {Gardner, John},
publisher = {Oxford University Press},
title = {From Personal Life to Private Law},
doi = {10.1093/oso/9780198818755.001.0001},
year = {2018},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
Lorem \autocite{sigfridsson,Gardner2018a}
\printbibliography
\end{document}
The output is the same.
addressfield: where do you put the publisher's address? Most people put it into thepublisherfield. Then it is difficult to split the information in this field. – gernot Mar 16 '21 at 09:46\clearlist{address}instead – Ivan Mar 16 '21 at 11:15\usepackage[notes,backend=biber,hyperref=false,giveninits=true]{biblatex-chicago}
\AtEveryBibitem{\clearfield{address}} \AtEveryCitekey{\clearfield{address}} \AtEveryBibitem{\clearfield{url}} \AtEveryCitekey{\clearfield{url}}
\addbibresource{/Users/fabrec/Dropbox/LATEX/texmf/bibtex/library.bib}
– Cecile Mar 16 '21 at 13:38