'Anyone' can ride any bike, but if you are miserable riding it you will soon take up another sport. Go for as small an inner chainring that you can get onto the existing bike and if that, with your new cassette won't get you up hills think about an 11-34 MTB cassette with a long cage dérailleur, or even a triple up front. Even with all of those items changed you won't be any more out of pocket than trading in a newish bike for another.
This is a very good reply, and worth taking notice of.
Some people take pride in being able to climb in a "big" gear (whatever your definition of that is). My partner is one of those, and I think she's nuts, to be honest. She has a triple with an 12-26 cassette on her road bike, and happily tells me the smallest gear she ever uses is 42 x 26 (42 being the size on her middle chain ring). She wouldn't be seen dead using the granny (smallest) chain ring.
I, on the other hand, have a triple with an 11-34 mountain bike cassette, and I use every single one of those gears. If I had to grind my way up hills, in my lowest gear, wondering if I was going to make it to the top, I'd be utterly miserable and probably give up cycling altogether. As it is, I can ride up practically anything short of a vertical cliff face.
So, while it's true that you are not cycling fit, and you will see a lot of gains in a short space of time to begin with, don't overlook the possibility that you might just enjoy cycling more if you have smaller gears.