Is there any app that does this and makes the whole archive searchable, of course?
Search for browser screen capture in the App Store and on Google. Here’s an example (that I haven’t used)
If you are a Google Drive user:
Google can index the content of most text-based files and certain encoded document formats. The most common file types we index include:
- Adobe Portable Document Format (.pdf)
- Adobe PostScript (.ps)
- Comma-Separated Values (.csv)
- Google Earth (.kml, .kmz)
- GPS eXchange Format (.gpx)
- Hancom Hanword (.hwp)
- HTML (.htm, .html, other file extensions)
- Microsoft Excel (.xls, .xlsx)
- Microsoft PowerPoint (.ppt, .pptx)
- Microsoft Word (.doc, .docx)
- OpenOffice presentation (.odp)
- OpenOffice spreadsheet (.ods)
- OpenOffice text (.odt)
- Rich Text Format (.rtf)
- Scalable Vector Graphics (.svg)
- TeX/LaTeX (.tex)
- Text (.txt, .text, other file extensions), including source code in common programming languages, such as:
- Basic source code (.bas)
- C/C++ source code (.c, .cc, .cpp, .cxx, .h, .hpp)
- C# source code (.cs)
- Java source code (.java)
- Perl source code (.pl)
- Python source code (.py)
- Wireless Markup Language (.wml, .wap)
- XML (.xml)
Google can also index the following media formats:
- Image formats: BMP, GIF, JPEG, PNG, WebP, and SVG
- Video formats: 3GP, 3G2, ASF, AVI, DivX, M2V, M3U, M3U8, M4V, MKV, MOV, MP4, MPEG, OGV, QVT, RAM, RM, VOB, WebM, WMV, and XAP
Rewind.ai From their website: AI generated notes of your entire day. Rewind is a personalized AI powered by everything you’ve seen, said, or heard. Your colleagues will wonder how you do it all.
I don’t personally use it so can’t vouch for it but it’s been getting a lot of buzz.
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I’m not sure what you mean by “searchable” (this app doesn’t archive content, only URLs), but I used HistoryHound for a few years: https://www.stclairsoft.com/HistoryHound/
I didn’t re-install it after I rebooted my Mac this year because I realised I’d only actually gone back to find a link older than 30 days (my Safari limit) a couple of times and figured I didn’t need it.
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Rewind.ai user here 
I’ve been using Rewind for a few months now and liking it more and more. It basically saves everything that you see on your screen to your local drive as well as providing the ability to record audio and automatically record meetings such as Zoom, Teams, etc. Worth checking out for sure.
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I have had success with History Book.
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Probably your national government. 
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