It's possible that they've been advised that long service awards could be seen as discriminatory and that's why they are no longer offered (it's what consultancies were advising people over 20 years ago, so it's not a new thing).
Unfortunately, SSP isn't that unusual; nor was it unusual for a small, family owned company to only start providing a company pension once auto-enrolment required it. Few companies offer true (valuable), Company-funded benefit packages anymore, despite what their job advertisements proclaim.
Really, if you're concerned about ill health, you should do more research or consider talking to a professional to make sure you're covered for what you most need. Private health insurance, critical illness insurance, income protection and life assurance are all separate but related protection benefits. PMI covers you for policy-specific acute (not chronic) conditions; CI pays out a tax-free lump sum benefit if you're diagnosed with a critical illness covered by the policy, and survive the date of diagnosis by a certain period of time; income protection pays a proportion of your salary for a set period of time if you're unable to work due to illness or injury (it usually only kicks in after a certain period of time - the expectation is company sick pay or SSP will be paid first then, if you fail to recover enough to return to work, IP will kick in). You can also get more specific cover, like cancer cover, which may be relevant to people with a certain age bracket or at risk group.
To answer your 'would you be annoyed with such a crap package' question: yes, I would, but I also acknowledge the reality that most Company-funded benefits are long gone and it's better to be employed with no benefits than have no salary or benefits.