For youtube.com, desktop and mobile devices receive different web pages. The page for desktop contains the OSDD(Open Search Description Document) <link>, and the page for mobile devices doesn't. That's the reason why youtube.com on iPhone(and Android I guess) is not added to the list.

Do following steps to check it:
1. Open Desktop Chrome;
2. Open Debug panel;
3. Change to mobile mode by clicking the button on the left top corner;
4. Open youtube.com;
5. Check the Settings>SearchEngines. The "youtube.com" is not added.

wolframalpha.com has the same issue as youtube.com.

For other web pages that don't have a OSDD <link> on both desktop and mobile versions, there might be differences between <form>s on desktop and on mobile, which prevents the page from being added to the list. wikipedia.org seems to be in this case, I'll check it later.

For the youtube.com case, you may want to consider filing an internal bug.

mpearson@ I've fired a bug for youtube.com (120535966).

For wikipedia, visiting it on mobile device will redirect you to the mobile version(e.g. https://en.m.wikipedia.org/wiki/Main_Page), which has too many sections in URL's path thus cannot generate a keyword. Code logic:
iOS:
https://cs.chromium.org/chromium/src/ios/chrome/browser/search_engines/search_engine_tab_helper.mm?rcl=7fa1000f175eeb5e0ee75287ab8c69feb13fdd2d&l=66
Other platforms:
https://cs.chromium.org/chromium/src/chrome/browser/ui/search_engines/search_engine_tab_helper.cc?rcl=7fa1000f175eeb5e0ee75287ab8c69feb13fdd2d&l=65

So wikipedia cannot be added as custom search engine on Desktop if you open debug window and switch to mobile mode. I've tested that.

If we want to fix it, we should fix it on all platforms to be consistent, and I think we should mark this bug as WontFix as we have not found any search engine website that can be added as custom search engine on Android but not on iOS.

+pkasting@ WDYT?

The original reason for the path constraint was the assumption that if a search engine has a path, the same host might well have multiple search engines, and we didn't want them warring with each other.

Perhaps this isn't always true and we should relax this constraint (everywhere).  I'm not sure.

I think this is a useful path limitation is likely a useful heuristic constraint.  I know that I often search help pages in retailer web sites, and don't want those clobbering the default search for items on the web site.  (Not sure if they would, just concerned.)

Issue 917365 has been merged into this issue.