I'm unfamiliar with your exact model but your description makes it very similar to the "King of Fans" branded models I fitted in my house twenty years ago and have fixed a few times. It might even be that yours are made by the same company as Home Depot source their Hampton Bay range from a variety of manufacturers including KoF.
You are entirely correct that you can access the pull chain once you have removed the lamp assembly. If it's just the chain which has snapped then it's normally straightforward to remove the short piece from the switch and install what's left onto the switch, but if your chain is the typical ball and link type and is more than a few years old then replace it with a new one if you can obtain the right size easily. They're not expensive, and the old one will have wear and tear making another failure highly likely. If either of the switches has failed (the pull switch or the fan direction switch) then they're also usually easy to obtain from any electronics store at low cost or you can contact Home Depot regarding spares
, also check here
. Your total repair cost should therefore be no more than a few £'s, $'s or whatever else you use locally: far cheaper and quicker than fitting a new fan.
I hope that my reply has been of use to you and saves you a lot of time and money. Please take a brief moment to rate my answer.