As a basic rule of thumb, I would suggest to go for monarchy first, then switch to republic "when you are ready", this means, you have some happiness control by HG, and Michelangelo (or maybe only temples) since martial law runs out, can afford to spend two foods for settlers, deal with away-unhappiness, can afford shield support for your units (maybe disband them, since they do no longer provide martial law).
I think in the early era, the advantages of monarchy outweigh the disadvantages, especially at deity: Martial law, settlers only eat once, three units without support vs less corruption and more trade arrows. You will catch up later.
This is a great and detailed answer Major Advantage
. I will add only that in my opinion in many situations including - and i would say especially - in deity as opposed to emperor, skipping Monarchy and going strait into Republic has substantial advantages of it's own along with the acknowledged setbacks which you so excellently described. i make this deity vs. emperor distinction because in deity with lots of cities we are forced to build tons of early warriors in Monarchy whereas in emperor there is a lot more breathing room. But even still, eventually when the city count reaches a certain point (which for me and many others happens very fast) tons of warriors are needed, even in emperor... instead of being able to focus more purely on settler building.
In Republic, the needed lux rate of maybe 40% (upon having a certain number of unhappiness triggering cities) is essentially free when we compare it to the limited number of arrows available in Monarchy, meaning that the value of 60% combined tax and science rate in Republic is comparable to the 100% combined tax and science rates in Monarchy. Also the civ saves on having to build dozens of warriors and can focus much more intently on building more fast settlers. Rep with HG and no buildings allows several cities to celebrate early to size 5 and Rep in Deity with about 36 critical mass cities can do this even without HG or any buildings... where decent trades can be made from several cities... and whereas in Monarchy once a certain number of cities are reached, the cities are generally forced to stay very small to satisfy the anarchists, even with HG (unless an expensive lux rate was to be employed which often seems in Monarchy to not be worth it).
Of course in a tiny map game with multiple nearby enemies (as in gotm 171) and with barbs to factor in (depending on barb setting), some games require us to have lots of defended cities as well as units away securing key mountains and hills, factors which provide a lot of additional weight to choosing Monarchy first. For this primary reason as well as because it was emperor level, i chose Monarchy first in gotm 171 in spite of being a great enthusiast of starting in Republic
.
i built a lot of very fast cities requiring a lot warriors as well as needing to keep these cities small (primarily size 2 max even with HG) with more settler building to keep the anarchists satisfied at zero percent lux. As one primary setback to starting in Monarchy, i felt it was not worth making a single "same land" size 2 or 3 trade, so i held back on trading until i could switch to Republic and celebrate to size 5s and 8's with HG and Mikes. This extremely low Monarchy trade value factor would be less of an obvious obstacle in a game with separate lands as well as the greater potential for longer distance trades... but no matter how you slice it, the trades in an early Republic are still far greater, in most cases twice as good or better which leads more quickly to an explosively thriving civ.
There are nuances to playing early Republic which can be embraced that make it work better. One of which i like to utilize at a certain point is allow most cities (with definite exceptions) to celebrate to size 4 or better before building settlers there... keeping them at least at size 3 so they are in a position to very quickly be restored through celebration growth to 4+ again before then building more settlers. But in the earliest stages, it is essential to rush settlers from nearly every size 2 city as fast as possible.
As one general rule, the smaller the map and the higher the barb setting, the more risky it is to start in Republic over Monarchy since we don't want to find ourselves in very early republic with a huge demand for lots of military units. Also i feel that Monarchy in emperor is a lot nicer than Monarchy in deity, whereas in Republic this setting doesn't make nearly as much difference. At a certain number of cities that would overload a civ in Monarchy, Rep works extremely well since anarchists which are immensely more responsive to the luxury rate are extremely helpful.
One additional important factor (among others) worthy of consideration is that Monarchy provides superior extortion potential from the ai, though this is not entirely absent in Republic through such actions as goading them into war by demanding they remove their troops and then saying no to peace and awaiting their offers to sweeten the deal with gold.
Overall i feel the choice of which initial government works best is not a clear black and white issue and is influenced by various factors, but i tend to favor Republic in most cases for my style of play
.