The list of services in your last 'graph are all what are generally considered
public goods, not in the sense that they are publicly
provided, but that they are
non-excludable and
non-rivalrous. That is, an ideal public good cannot limit who benefits by it (directly or indirectly), and additional users don't affect total availability (much). These are both ideal conditions, and there may be
some degree of excludability and/or rivalrousness in practice, but generally public goods have far less as a matter of
degree.
Other notable public goods are national defence, information, other general infrastructure, communications technology, etc.
There's another class of goods and services which are often at least in part government-provided because of coordination problems when provided by independent entities. Examples would include standards (weights and measures, specifications, safety requirements, shipping containers), transactional interchange formats (e.g., billing and purchase codes), large-scale information gathering (census, economic data), and the establishment of laws, regulations, courts, and enforcement themselves. Max Weber's definition of government as having a monopoly on the claim to legitimate use of force is another such case: regions in which that legitimate claim monopoly doesn't exist, in any of its three variants (not a monopoly, illegitimate, ineffectual force) are effectively ungoverned.
And then there's the case of insurance services, especially over broad areas or long periods. Pensions, healthcare, and disaster insurance would be three areas in which governments often, though not always, play a role, largely because private provision of such services simply fails to meet social needs.
If one looks at national governmental budgets, the largest items are typically pensions (e.g., Social Security in the US), healthcare (Medicare), and defence, with all other spending (largely: collective-action regulatory roles) being about a quarter of the overall total.