This is a long answer, so get your popcorn...
1) Yes, IGE works. Its always worked. The problem is that you've entered the world of Mods and you don't know how to properly live in it. But that's OK, everyone had to learn how including me. Keep on reading.
2)Yes, sometimes (depending on Mod/Mods) the IGE button will not show. This usually happens in two instances: the very first turn of a game, and if a Mod puts so much crap in the status bar that the IGE button is simply pushed off of it. Keep on reading.

How to live in the world of Mods:
Make no mistake, IGE is a Mod like any other Mod in that it adds and changes the code that makes civ5 work. IGE was 100% compatible way back when it was first written but that was 15 or so years ago (I think). Since then, hundreds of Mods (and maybe even some DLC from Firaxis but not sure) have been created that IGE was NOT made for. For me and the Mods I use, IGE usually played nicely with them but that is NOT going to be the case with all Mods. There ARE Mods that make IGE only partially compatible (such as Gaia) and I would not be surprised if there was a Mod or two that are totally incompatible. Totally as in...you crash right off.

Is the Mod incompatibility fixable? Of course, everything is fixable because these are all just computer programs. But the author is not going to do it because he doesn't do this stuff anymore. So either you or someone else will have to spend a few hundred hours learning the LUA language and then create a new IGE version. Either that, or you'll just have to pick and chose which Mods will work the way you want with IGE. Is there a list for this? Not that I've seen. So, welcome to the world Mods. 8-)

The IGE button:
The way to get around a missing IGE button (assuming your other Mods allow IGE to work) is to use the keyboard shortcut, CNTL I. That's "I" as in the 9th letter of the English alphabet. But be aware that in order to change the options of IGE, you need that button so that you can right click on it to get the drop down options window.

General problems:
I see a lot of complaints about "not working" that appear to be simple cache problems. Keep in mind that civ5 was NEVER intended to have a bunch of Mods in it all fighting each other for their existence. When you enter the world of Mods, you leave the world of normal play and therefore leave the normal programming parameters behind. A common occurrence during this is civ5 creating a cache that is either corrupted or incompatible. This can happen live or can happen during a savegame. If you're lucky enough, this can be your present problem. I say lucky because its an easy fix: delete the two cache folders. The game will auto create them as necessary.

I can't say where civ5 puts your cache folders because different OS installs can put them in similar but different places. I can only say that MY install, in my Win11 Pro, puts them in:
c:>Users>(your username)>OneDrive>Documents>My Games>Sid Meier's Civilization 5.

Note that in Win11, there are like half a dozen folders named Documents (because Microsoft does not believe in doing things in an easy or user friendly manner). THIS particular Documents folder is NOT the usual yellow one..its the grey-colored Documents folder that OneDrive uses (because in Win11, Microsoft wants to shove OneDrive down everyone's throat).

There are two cache folders, named Cache and ModUserData. Simply shutdown civ5, delete those two folders and restart the game.

One last thing...some IGE incompatibilities can be avoided by not running IGE as a normal part of your game. Simply play your game without it until you need to do an edit. Then save the game, shutdown civ5, then restart civ5 and include IGE. Then do the edits you want to do. Then save using a different savegame name (in case this method ends up failing) and then shutdown civ5 again. Yes, AGAIN. Then delete the two cache files, restart civ5 and this time untick the IGE mod. Is this method long and boring and a pain in the ass? Yep. But in the world of Mods, this is the price you sometimes have to pay if you want to play your way.

Have fun. 8-)