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No they don't, it is the exact same ratio. The benefit OP is asking for is in space savings.
They process 100 times as many materials at a time but take up only about 3ish times the space. I have not timed them to see what their speed efficiency is but it does seem to be at least even if not faster on a per material basis. However I did notice that they do not benefit from the sleeping bonus, or if they do it only applies if you're afk for a very long time as it might be per craft rather than per product as it implies.
As for the space issue you mention op, I can tell you with certainty that I use WAY less space for them than before that. I was only able to fit in about 10~ of each type of processor. Now I use 3 of each mega for copper, iron, and steel and that produces 3 times as much for about half the previous space. This is despite also included space for packagers and other extra small buildings. The advantages are plenty clear.
They do seem to be faster. 3 megas for copper, iron and steel gave me slightly more steel than 6+ of the regular ones. They do take long to get running because of the whole packing thing though, and I've found that it doesn't seem to be faster nor more efficient for coal.
Also
They have an internal priority whereas they won't fill containers if they have recipes to do. If you also have packagers and production buildings in there, they will not fill the rocket until every packager is running and every furnace/assembler/refinery is fed. You can try to toggle their priorities in the antenna and have one logistics bot with only "deposit resources" as directive.
Until you unlock the blueprints at the station they're not quite as useful.
I'd advise limiting the packager output to a single crate, especially for the boxes going to rockets. And for rockets you probably want to manually drop resources in for packaging so as to avoid having a box of stuff with nowhere to send it.
Sure they have a little better ratio but for example on an island with copper and iron I tied them. One bot gathering Iron one copper, so three logistic bots. With all the other stuff going on on that island the packaging took so much actions that it completely overwhelmed the logistic bots. They did not even have actions left to send the steel of the end of the chain away to another island that needs it.
So basically I think they are a design error. Or the logistic bots are too limited.
Maybe packers need something like their own logistic bots so the island wide ones are not overwhelmed with it.
I will say I held off really getting into using them until after I got the skill to speed up the process but generally once they're running there's not really a problem.
I'm also kinda low key getting vibes that the big furnaces aren't receiving the gem bonus, but no way to certify.
I did the numbers VERY roughly with full upgrades and diamond gem:
The Advanced Electric Furnace took about 1m40s ~ 100s to produce the steel recipe. The Mega Electric Furnace took about 8m 20s ~ 500s (a bit more I think). Advanced produces 3 steel per recipe, so about 0.03 steel per second (33.3s per steel). Mega produces 75 steel per recipe, so about 1 steel every 0.15 seconds (6.7s per steel). So the Mega is about as fast as 5 Advanced.
Mega takes only 5x5 = 25 tiles, while 5 Advanced take 5x3x3=5x9=45 tiles. You need to keep in mind that 3x3 tiled packers are needed for the Mega, at least 2 for the recipe, meaning 25+18 = 43 tiles. You probably don't need more packers if you add more Megas though.
Mega needs 8 power. Advanced 5x2 = 10 power. Packers don't need power.
So Mega is always better.
I haven't checked with the resting bonus. This could be an issue.