I'm not a linguist but I do know both languages (English and Spanish) fairly well, and my impression is that this is mostly a semantic argument, which is based around the way we conjugate a single word.
If you were very strict about only looking at the single word, English stays very simple (do=infinitive/present plural, does=present singular, doing=progressive, did=past, done=arguably not a tense because it's an adjective). If you look at it in this way, Spanish obviously has more tenses--they have six different variations of just the present tense, for example.
But that doesn't capture the whole story because English does have all those tenses, English just doesn't do it in one word. Notably, there's no future tense in the one-word variations, so nobody reasonably will argue that the above is all the tenses.
And that takes you to a complicated place because you can do multi-word variations in Spanish, too, and sometimes those are functionally the same (what's the difference between "voy a pagar" and "pagaré"?).
And if you start allowing for multiple-word tenses, then there's a virtually unlimited number of constructions possible, and any limitation you could place on this is completely subjective. For example, your link doesn't list "I will have had been doing" as a tense, but it's obviously a grammatically correct construction with a real use--the only reason for not including it is a subjective judgment that it's too uncommon.
And critically, you can do all of those in both languages, it's just a matter of how many words and how awkward/uncommon the construction will be. The different structures of the languages lead one toward preferring certain structures ("pagaré" over "voy a pagar" in Spanish, while one would likely say "I'll pay" in English because there's no one-word option). But to represent these as different tenses is pretty subjective and maybe a bit silly.
The one thing I'm aware of that Spanish can do that English can't is communicate formality with a conjugation, i.e. "tú" form versus "usted" form, but I'd argue that's not a tense, because "tense" seems to be about the the temporatlity of the verb and that's a modification to accommodate the formality of the subject of the sentence.