Short Answer
The subject of the second sentence is es and of the first Arbeitsplätze. Thus the main verbs do conform with the corresponding subjects.
The following deals with explaining the presence of es in both cases. They may be viewed as representing two very different phenomena.
Second sentence
This is just a variation of
Es geht jemandem gut. / Someone is well.
You can regard the es here as fulfilling the need for a subject in the sentence. Similar to English
It rains.
Or even better the it in:
How is it going?
You notice that in the first highlighted sentence es does not care about case and number of jemandem. Rather, es is a placeholder, like it in the English sentence. Therefore you may say:
Es geht Papa gut (masculine)
Es geht Mama gut (feminine)
Es wird den Bürgern besser gehen (neuter plural)
*Es* is the *sentence subject* placeholder and doesn’t *refer* or
*point* to *Papa/Mama/Bürgern* in the above sentences!
Nevertheless, *es* is the sentence subject to which the verb in each
sentence — *geht/geht/wird* — conforms.
What I explained here for es geht jemandem gut similarly holds for es gibt, es ist kalt,...
Es gibt keine Drachen. / There are no dragons.
Es war der Jungfer kalt./ The miss was feeling cold.
First sentence
By permuting the parts of the sentence while omitting es you can see that es is not explained likewise here (i.e. it is not replacing a missing, grammatically required subject). You could permute it into a legitimate sentence without es in the following way:
Viele neue Arbeitsplätze werden entstehen.
This straightforward sentence order usually emphasises viele. Your sentence order was chosen to convey a different emphasis (it could be an item in an enumeration of clauses). But why does it contain es?
Foreign language learners books on German sentence structure and Wikipedia on German sentence order love to include a rule similar to this one:
The conjugated verb is the second element in the main clause and the last element in the subordinate clause.
Teachers using this rule will say the es is there so that werden is in the second place. They will then teach you to view “viele neue Arbeitsplätze” as one chunk (subject and modifiers), so that their rule also covers:
Viele neue Arbeitsplätze werden entstehen.
*Es* makes sure the sentence structure is idiomatic
and does not *point* to Arbeitsplätze!
*Arbeitsplätze* in turn is the sentence subject,
to which werden conforms.