I am a native speaker and my answer is not a guess.
When he talks of "enemies of the people", the analogies with Stalinist Russia and other 20th-century regimes are so glaring that you have to keep reminding yourself that the French Revolutionaries were having to make everything up as they went along, against a very different background from that of the 20th-century tyrannies.
The verb in question is not "have" but "have to." And yes this verb works very much like "must." For example, instead of saying
I must admit that I have never come across this form.
you could say
I have to admit that I have never come across this form.
The difference between must and have to is that in the former, the obligation of the speaker is imposed, whereas in the latter, some other obligation is imposed. See “I have to” vs. “I must”.
And yes, "were having to" is a progressive form of "have to." Namely it is the past progressive. The present progressive is "am having to."
The use of the past progressive here (were having to) is not required by either (a) "as they went along," (b) anything else in the near context, including other uses of the progressive, such as "are so glaring" and "have to keep reminding yourself." And yes, you could write the sentence as:
When he talks of "enemies of the people", the analogies with Stalinist Russia and other 20th-century regimes are so glaring that you have to keep reminding yourself that the French Revolutionaries had to make everything up as they went along, against a very different background from that of the 20th-century tyrannies.
However, by changing the aspect you are changing the meaning (e.g., what is expressed by the aspect). In this case, the original is using the past progressive and your version uses the past simple. Either one is fine; again, nothing in the context requires that either aspect be used. It is the author alone here that prefers the progressive, much to the detriment of his prose.
P.S. And what about this swap of aspect:
"Revolutionaries had to make everything up as they were going along".
Yes, that is acceptable. All that does is stretch out the activity of they went along (i.e., you have another example of the normal difference between the simple past and the progressive past).