I honestly think you could swing a few votes by bringing up how pissed Hillary would be by not being the first woman to be president.

Basically, yea. The Democratic Party makes the rules for how they pick candidates, and this is them following those rules.

Parties could choose their candidates through Ouija boards and it'd have been fine as far as the law is concerned

Yes, pretty much. We didn't even have primaries until the 20th century. Before that, the party would just pick the candidate at the convention. Even then, until 1968, the primaries were basically just opinion polls, and party bosses were free to ignore the primary results. In the 70s, they started forcing delegates to commit to primary voters' choices, but that's simply an internal party rule, and they could change it at any time. Also, even now, the party has a lot of control over who is nominated. The Superdelegates are not committed to voters' choices, and in 2016, they were the reason the AP controversially called the primary for Hillary just before California voted. The fact that primaries take place over several weeks instead of a single day, like a general election, also gives the party time to place their thumb on the scale for their preferred candidates (something that Representatives Ford and Smith recently admitted the party did in 2020 to give Biden the nomination).

So, tl;dr: yes, the parties can do whatever they want. Until about 50 years ago, the primaries were basically just suggestions, and even now, they party is doing more to select the candidates than you realize.

Yes, parties pick their own candidates and there's no rule saying we actually have to have a primary.
The parties could just put forward whatever candidate they want and push them for president, but if they do a primary they have a better chance at winning because they can select the most popular candidate with voters.
The primary is there for their benefit, not ours.

In the UK, generally chosen by party membership. There's been some experiments with open primaries, but nothing really substantial.

It's probably worth mentioning that, because the timings of our elections are generally left to the whim of the Prime Minister, candidates are normally elected by the party way in advance so they're ready just in case anything happens. Our election cycles also usually last only six weeks, which isn't enough time to run an internal election and then campaign.

In Germany, the president is elected by a group of politicians and public figures, not the public. But the president's duties are mostly ceremonial.
The chancellor, who is head of government, is elected by the members of parlament, right after the parlamentary elections.
It's kind of a public election, because the party with the most seats in parlament gets to pick one of their members for chancellor, and that choice is made public before the elections. They announce a "chancellor candidate" well in advance.
In no election does the public get to vote on any candidates before the actual election. They're put up by the parties in any way they choose.

Well, in Sweden the party often chooses but you can also vote for a particular person if you'd like. It's not mandatory though. This is for all levels, country wide, county and municipal.

This is going to massively depend on which country you live in, but frequently neither.

Parties can pick who they like, but they often allow politicians and party members to vote as part of internal selection process.

In the UK only weirdos and political extremists are party members, and the Tory party tends to spend a lot of effort trying to stop their members from having a vote.

So of the last four prime ministers.

Sunak didn't have a vote (lost to truss before that).

Truss won an internal vote.

Johnson won an internal vote.

May was uncontested.

And this is only the internal vote. All of them became prime minister without an election. Generally you vote for a party (some pedant will claim you vote for MPs, but they do what the party says) and then the leader can change while they're in power.