The biothane loops are just to carry your traces, keep it tidy or prevent a foot from getting over in a bucking situation.
I think there may be some terminology trouble in this conversation.
In the top picture- those straps wrapped around the shafts are not tugs- they are the hold backs, or britching straps.
There are two schools of thought on the safety of connecting with snaps- (leaving them on the cart or going with the horse)
One is that snaps wear, break and lose their springs- and the last thing you want to do is loose your britching function because a snap broke. This camp says- “Never ever connect with snaps”
The other side says- “In case of emergency- being able to quickly unsnap your horse can save critical time… I feel safer with quick release snaps”
A third corner suggests: The back strap can connect to your saddle with a snap- and in an emergency you can - with ONE snap- release the entire britching and avoid the problem of having to go to both sides of the horse (or if the horse is laying on top of a shaft)
My personal experience with that idea- I purchased a horse and harness from someone who did it this way- and discovered that the snap at the backpad was worn almost all the way through- so that when I examined it- I was able to break the snap with my bare hand. While the idea seems great- I don’t like the idea of metal to metal connections that are not fastened and unfastened by hand every time you hitch… metal to metal wears in a way metal to leather doesn’t- and if you aren’t paying attention it could take you by surprise.
The cart is adorable and your pony is a cutie!!
I will second what Renae said about Bridles and hitching safety protocol. Just get out of the habit now. Don’t do it… if you were to do that in public- at worst it could get you, your pony or someone else seriously hurt- lesser consequences might get you excused from the grounds… yelled at… lectured, politely instructed or socially ruined for life… just. don’t. do. it. 
This form of warning on the internet is the most painless - ((hugs))
It actually looks like the whole harness is a bit too small and undersized for your pony- the back pad seems very thin for supporting a cart- and the britching seat also seems very thin- not providing much surface for her to brace against.
One last note about how to attach the hold backs to the cart- there is a certain way to do it, and from the looks of things- your straps aren’t long enough to do it right. The footman loops are intended to hold the position of the hold backs secure on the shafts- but they are not really intended to hold back the whole cart- (as they are only attached with a few tiny screws) So the hold backs should be wrapped at least once- maybe more around the shafts. Also- if you keep the hold backs with the horse harness- they aren’t just fed through the ring- but are secured at the ring doubling back on themselves and locked in with a keeper. (I can’t see the picture clearly enough to know if this is how you have it) When you wrap the shafts- you begin with the strap between the horse and the shaft- you feed it through the footman loop and then over the shaft- wrapping forward- after the correct amount of wraps- the loose end is tucked (locked) under one of the wraps before it is finally buckled into adjustment. This style of wrapping assures you the most points of “sticking” were a part of the leather to break- it wouldn’t just all flap to the breeze instantly- youn might have a little more chance to get stopped before it all came undone.
