Intuition behind SEM latent variables not being actual variables There are essentially two ways, to my knowledge, to put together a number of numerical items that are all meant to quantify one abstract notion.
One can average the items and create a composite score. This can be done in a more or less sophisticated manner, e.g. using weights given by Principal Components Analysis.
One can use Structural Equation Modeling and create a latent variable.
I tend to think of a composite score and a latent variable in the same way, to me they achieve the same job. Yet the latter does not seem to give you an actual variable with an  individual score for each subject in a study, a tangible variable for which you could provide descriptive statistics.$^{(*)}$
What is an intuitive explanation for that?
$^{(*)}$In fact, the only way to use a SEM latent variable, for instance as part of a statistical model, is within the scope of SEM itself. It is known, by the way, that composite scores should not be used in SEM analysis, and replaced by latent variables.
 A: It is not uncommon to estimate an SEM from which the values of the latent variables are computed, and to use these for some other modelling (SEM or otherwise). Most SEM software that I am aware of allows the outputting of the latent variables.
The intuition behind latent variables being "actual" variables, as per the question title, is simply that they are unobserved. This is the literal meaning of a latent variable.
As for the ways composite variables are computed (sums or averaged) or in other cases such as PCA or exploratory factor analysis, we think of these variables as being computed from the raw variables and thus caused by them - so in a diagram you would have arrows pointing from the raw variables to the composite variable. On the other hand in SEM and confirmatory factor analysis we think of the variables as representing something that is unmeasured. A good example is "true" blood pressure. We can measure a person's blood pressure at various times during the day, and there will be natural biological variation in these measurements. There may also be variation due to measurement error. But from a clinicians point of view, they just want to know the "true" blood pressure (for example does the patient suffer from generally high blood pressure). In an SEM, we would have arrows pointing from the latent variable to the raw variables, because we think of the raw variables as being caused by the latent variable (plus some error).
A: I am not entirely sure if I follow your terminology. A SEM involves two parts: a measurement model and a structural model. If there is no measurement model then the method is called Path Analysis. If there is no structural model then in most cases it is just Factor Analysis.
One of the most popular ways to set up a measurement model of an unobserved construct is through the latent common factor. Thus, Confirmatory Factor Analysis (CFA) is an example of an analysis that uses a common latent factor but does not require a "SEM analysis".
You are right in seeing composite scores and the common factor model as equivalent, as the first is just the latter with specific constraints and assumptions. One of these assumptions concerns the difference between formative indicators and reflexive indicators. In the first case, indicators form the underlying construct, i.e. they entirely determine it. In the second case, the indicators reflect an unobserved variable that causes them. Statistically speaking, this is usually associated with the assumption regarding the measurement of the construct: in the first case indicators have no measurement error, in the latter there is measurement error assumed.
Regarding your example, using the weights of a PCA to compute a weighted average as a score would correspond to a formative construct: the resulting score is fully determined by the indicators, and not the other way round.
When dealing with reflective indicators, in lavaan you can use the function lavPredict() if you want to estimate latent factor scores for each subject. Note the emphasis, as this is a prediction involving error: the indicators are used to imperfectly measure the underlying latent variable that is assumed to cause them.
