When I tried https://www.gov.uk/apply-online-to-replace-a-driving-license to obtain a replacement UK licence, I ran aground straight away as I am not living in the UK. Is there any way I can get a replacement under my current circumstances?
Drivers licenses are issued to residents of a country, or US state, or Canadian province, or whatever entity or political subdivision is doing the issuing. Issuance requires proof of residence.
The same reasoning applies to renewal: residents only. While some US states allow by-mail or online renewal (New York state comes to mind), the licensee must provide a within-the-jurisdiction address to qualify. Licensing authorities will therefore decline to replace a non-resident's previously-issued license. This is what stopped you on the gov.uk website.
If you use someone else's address to renew, the assertion in the application that you are a resident will be false. If revealed, this falsehood will result in the cancellation of any license or renewal so issued, and may imperil insurance coverage obtained by or used in concert with that license.