I am assuming you mean the sidecars sold in the USA, with the "COZY" Brand Name. I am a long time side car driver for 25 + years, so I will give you my opinion.
All the COZY's are "LIGHT WEIGHT" Sidecars. I have found that the HEAVY sidecars, with a larger displacement bike designed to pull them, to be the Best, safest, & most enjoyable to pilot. Outfits like BMW/STEIB, ZUNDAP/STEIB, URAL, DNEPR, Chinese CJ, Do function best because they were designed to be used with a sidecar. They all have Welded in Frame eyelets, & front ball attachments provisions, they all have HEAVY FLYWHEELS & DRIVE SHAFTS (suited to pulling) All these outfits are as safe as a Sidecar can be, & engineered specifically to do the job.
WHEREAS: The lightweight accessory sidecars (COZY) are universal mount. this frequently translates to U-BOLT to frame attachment points (U-BOLTS can move or dent the frame). The UCE Enfields Aren't really suitable, because they do not have a heavy full loop frame, they have a lightweight frame, that uses the engine a structural member. The Flywheels are somewhat lighter than what I would want to pull a "Hack". Also keep in mind the fact that pulling a hack places more of a burden on chain, Clutch-Drive Train, Brakes, Suspension, & if Improperly aligned will eat Tires like Potato Chips.
Bikes like HARLEY DAVIDSON, PANTHER, & JAWA are the only CHAIN DRIVEN bikes suited for side car tugs, that I can think of. You have probably heard the expression "use the right tool for the job" This is especially true when considering a sidecar outfit.
You would be much more satisfied with a URAL. If you are mechanically inclined, they are easily owner maintained & super tough, they are reasonable on the USED market. Lots of folks buy them then decide they don't like sidecars. My first one was a 1980 Dnepr. My last one was a 1994 Ural Tourist. I owned it 7 years & loved it. but found when I moved to the mountains, it was to dangerous in the traffic on mountain roads. Sold it & bought my 2010 Royal Enfield G5 Deluxe. I would NEVER use it as a side car tug. Pictures of my DNEPR & URAL outfits below.