It seems like you have an interesting definition of liberty. Liberty (to me) is freedom from authority. Libertarians core value is not having government force individuals to do anything. If people want to opt into a universal healthcare private system they are free to do so (kind of like insurance). A big motivation for this is lack of trust in government to handle the job well. Libertarians see government as inherently prone to corruption and thus want to limit their power as much as possible. The extent to which a given libertarian wants to limit government varies. By appointing government authorities to the system the cost of everything rises as in addition to health care you also have to pay the government workers who oversee the system and it's not very efficient. Not to mention politicians get to decide how much money goes to these programs etc etc. do you really want politicians involved in your health? With all the inefficiency and corruption in politics why do you trust them to handle your health?
For a bit of context. Inviting Trump was decided on by the party leaders and was not a popular choice with most of the delegates who were worried it would send a confusing signals to people not familiar with the party. Trump was also able to sneak a lot of his MAGA supporters into the convention for the speech. They tried to sit up front to make it look like Trump was popular there but they were forced to sit in the back. So any claims about cheering should be take with some salt.