The evidence is overwhelmingly against you[1], given how the plastic bag tax in Ireland and now the UK has shown it does significantly increase reuse, even for nominal amounts like a 5p charge per bag.
You might not want to save money, but most consumers do.
You're falling into the trap of "everyone is exactly like me", where you're in fact the exception, not the norm.
I think bags are quite different: the hassle of reusable bags is really low, they weigh almost nothing, and they have other real advantages over the plastic bags (they don't break).
Many US states have bottle deposits for a similar "nominal" amount per bottle and the majority of people don't do it: when I was a broke college student some friends and I did it for a little while but the math really didn't work out even then.
My Aunt & Uncle and in Holland did it all the time and as far as I could tell everyone did it (I'm from UK, we don't have a scheme like that).
In this article after a very quick google they report a 90% return rate in Holland (I'm not going to spend time digging into where the 90% comes from):
So it might be a cultural thing or it may be the deposit figure is too low. As far as I remember in Holland it was 10 cents when I was a kid in the 90s, which would be about a dime per bottle.
It certainly is cultural: in Switzerland the rate of recycling is extremely high without and deposit.
To clarify, in my experience what happens in the US is most people recycle bottles at curbside pickup, rather than drive somewhere to get 5 cents per can. In cities homeless tend to go around and collect all of the cans from the curb and redeem them.
This is not only cultural and has nothing to do with the Swiss being more ethical or better people.
As always, in Switzerland the system is very smart: The main reason for the high recycling rate in Switzerland is that a 35 liter trash bag costs around 80 cents. The cost for the garbagemen etc. is included in the purchase price of the trash-bag. (Also, the trash-bag is of high-quality and never rips, so garbagemen who get a >4000 CHF monthly salary, almost never waste time picking up trash from ripped trash-bags.)
This is unbelievably smart because this nudges people recycle everything: Paper, carton, glas, metal, and plastic bottles are all thrown away in extra containers that are distributed conveniently within the neighbourhoods. People really fill the expensive, high-quality trash-bags mostly with non-recyclable stuff. If you put in recyclable stuff there is no punishment or anything, you just have to buy more bags, which hurts your pocket.
Compare that to Germany where they rolled out a machinary of expensive plastic-bottle-bar-code reading robots in each and every supermarket that had to be engineered, maintained and protected against tampering/hacking/fraud with false bottles (from Eastern Europe).
You might not want to save money, but most consumers do.
You're falling into the trap of "everyone is exactly like me", where you're in fact the exception, not the norm.
[1]https://www.theguardian.com/environment/2016/jul/30/england-...