It's an interesting choice to suggest that the switch to Hepburn romanisation was motivated for a desire to better help English speakers pronounce Japanese words when tsunami is your example. The official Kunrei-shiki romanisation for つなみ is 'tunami', and I can promise you that nobody who visits Japan tells their friends and family that they visited Mount Huzi (ふじ). You would have a point if you had chosen something like Mitutoyo, but even then names are usually the exception when it comes to romanisation/anglicisation as official rules are less applicable, cf. Mitsubishi.

Still, something like 'sooonami' is particularly grating even if we ignore the pretentious BBC accent (I have heard tsu-na-mi on BBC shows to be fair). It could be because as you said the onset gets simplified to better fit English phonotactics like with other words: (ph)thalic acid, (p)terodactyl, kr(w)asan (croissant) in American English with a doubly 'wrong' t at the end, (k)nife, (g)nome, sometimes (g)nu, etc, but I don't think this is it. Su-na-mi sounds fine and this is how it's pronounced in Spanish and some other languages too, every language ends up 'mispronouncing' words if it doesn't fit nicely into the existing phonology. I think what bothers me the most about 'sooonami' is the stress inevitably gets placed on the second syllable which becomes 'nah' in non-rhotic accents which just sounds wrong, and in terms of Japanese phonology it's rare to place the stress on the middle syllable, never mind that the mora is wrong and the pitch accent is wrong, but I by no means speak Japanese.

As for why English even uses tsunami in the first place, maybe 'tidal wave' makes sense if that's what you grew up with or you live in a part of the world at risk of tsunamis, but I don't think I made the connection until I was an adult. Are all tides not waves? Tidal bore, tidal flood, storm wave, etc, sure, unusual events relating to the tide or weather, tidal wave fits if we ignore that they're not caused by the tide, but it doesn't seem comparable to me even if tidal wave isn't wrong and is synonymous.