I have seen that when I capture (with a chrome extension or using safari) a web page, if this web page has a popup with a cookie policy, it seems that it does not capture the web properly …
Per-application cookies aren’t shared (and haven’t been for quit some time in macOS).
Add a bookmark to the site and log into it in DEVONthink. This will set a cookie for DEVONthink and potentially allow avoiding this behavior (and note I said ‘potentially’).
I don’t think there is a way to directly avoid this; you may find that an ad-blocker of your choice (or pihole) can be used to avoid seeing these popups and that capture may then work better.
You may also find that different methods of capture (HTML, PDF, Markdown…) work differently (and better or worse) for different websites - it’s worth experimenting.
(Edit: sorry, I responded at the same time as Jim; please don’t understand my post as contradicting his, but as an adjunct to his post).
It is that I am comparing with Evernote to see the behavior of one and the other and for example Evernote seems that it makes the captures in a better way, it seems that it extracts them correctly.
Anyway, he couldn’t understand very well what you’re saying (add it as a bookmark and log in to devonthink, I mean).
In Safari, go to the site. Tap the share icon. Select add to DEVONthink 3. Select Bookmark as the format. Now in DEVONthink select the (newly added) bookmark. DEVONthink will open the webpage in the preview area in DEVONthink. Now you can select whatever response you want in the cookie popup. That way a cookie will be stored in DEVONthink’s browser; now when you capture a page from that website, you may no longer capture the cookie popup.
It seems that I have followed your steps but it still takes the capture wrong, I sense that with devonthink unlike evernote, it has a different way of capturing html webs and in that, it fails.
The method is almost certainly different; capture works very well for some pages, less well for others. You could try printing the webpage (or better the Reader View) to DEVONthink (so in Safari do ⇧⌘R and then ⌘P and select PDF/Save to DEVONthink 3). If that works for you, you can add a shortcut to the last step, so that ⇧⌘R ⌘P ⌘P would work)