The reCAPTCHA module, when used on the Contact Us page, works perfectly for us and always has done. It's possibly a little out of date now (V2 when v3 is the most current release) so this
might explain why the link url provided within XC5, gives a 404 error (see below for a link that does work) but V2 still has a very high, common use figure outside XC anyway...
Having said that
We think the default XC5 "Contact Us' page (not the reCAPTCHA module itself) is very poor and we covered this is detail, a long time ago now
HERE 
Your signature says
XC 5.1.10 is that still correct? It seems at odds with the 'latest updates' you mention in your post, unless they refer to reCAPTCHA data updates
That question ^^ is because there's more serious issues than various reCAPTCHA errors, in those older XC5 releases
FWIW This url link does work (currently)
https://www.google.com/recaptcha/admin#list You can either modify and/or add new reCAPTCHA installations c/w all the required verification data fields. We (luckily) get minimal spam on XC5 probably because of reCAPTCHA yes, but, more likely, because we block lots of IP addresses at server level, so they never get to XC5 in the first place (Russia, China and Brazil currently seem to be the worst...) That option might work for you too if you have server root access etc