Moras may form the base of Japanese but syllables play a part as well.
For the video in question as stated in the previous answer, in the second video there are two separate moras. Even if I clapped out the "syllables" it would probably sound like: ko-i-zu-mi-kun. This could be in part to the pitch accent falling on the い.
And on clapping syllables here's a video from a Japanese linguist Shigeto Kawahara where he demonstrates how his five year old child will clap mostly based on the syllable: https://www.youtube.com/watch?v=jtOju8g2FSI. It also has an explainer on syllables in Japanese. TLDR for that: There are special moras: っ (consonant gemination), ん (moraic nasal), ー (elongation) and occasionally others. Japanese syllables are made up of heavy syllables and light syllables. Heavy syllables have a construction of
normal mora + special mora. Light syllables include all normal moras. There is a connection to pitch accent in that: you can't place the accent on a special mora, which means you can only place the accent on the first mora of a syllable. If you apply this to こいずみ, since the pitch falls on い, then the こ and い must be separate syllables.
To address your bottom questions:
So for those consecutive vowels things, do you really have to say them together?
These kana + smaller kana combinations could collectively be called 拗音 (youon). There isn't really any separation to begin with, so yes you have to say them together. It helps people learning, to explain the pronunciation as: quickly saying these as two separate moras but in terms of the actual sound: when ゃゅょ are added, you end up using the consonant from the original mora and the vowel from the small mora and merging them. The consonant additionally becomes palatized which makes it sound like you are gliding between vowels when you are in fact just pronouncing the consonant at a higher position. Examples of this: きゃ, ぎょ, びゅ.
Also the problem gets more complicated (or easier in another sense) because the 拗音 you picked (しゃ) actually doesn't get palatized like others. It's actually the same sound as if you changed just the vowel in し to an あ or whatever that smaller mora's vowel is. This is true for し (しゃ, しゅ, しょ) じ (じゃ, じゅ, じょ) and ち (ちゃ, ちゅ, ちょ). So when you learn あいうえお you can just add the し consonant to produce しゃ, し, しゅ, しぇ, しょ.
Why is there like a special thing ゃ for しゃしん but not really for consecutive vowels eg a 'lowercase' い for 'こい' ?
The first thing (拗音 youon) is a single mora formed of a consonant (sometimes palatized) and a vowel. The second thing is two separate vowels that form two separate moras. So the duration of the second would be twice as long. Hence why the 拗音 isn't just two full size separate kana. Technically speaking, the timing of two consecutive vowels like あい should be the same length as any other two moras but in fast speech this isn't the case. As such linguists like Kawahara refer to these as one syllable. But as you've noticed they don't always act as one. There are a couple influencing factors like speed of speech, pitch accent, idiolects etc...