I don't think it would affect the traditional presidential succesion (two terms are enough)
Oh, it definitely would. The whole 'Cincinnatus' vibe of serving a period and then stepping down was a HUGE precedent and support for stable politics in the US. If he died in office, revisionists would immediately say 'Well, yes, he SAID he'd not run for a third term, but surely with the French crisis, he'd have seen we needed a steady hand on the tiller.'
No, I think without that specific example, you'll have people running for third terms on the slightest excuse.