Designing an IoT product
Size &amp; Components

I'd say that if you're interested in minimizing size, choosing to use a WiFi board such as the ESP8266 along with the Arduino's large chip (Assuming you mean DIP form) isn't the most space efficient route, but I think you've figured that out.

If you want to continue with the Arduino's chip for ease of programming you could get it in a smaller, possibly SMD, format, and find a suitable WiFi module to accompany it. One that I found with 2 min of searching is the ESP8266 ESP-01, which seems to work with Arduino, as I found a page of projects here

The ESP8266 (Full board version) (or at least the one I have beside me) is 60mm x 31.5mm which is only slightly smaller than the Raspberry Pi Zero W at only 66.0mm x 30.5mm x 5.0mm. I assume your initial rejection of the Raspberry Pi didn't consider the multiple forms it comes in. It's only $10 and would allow a great suite of programming options with a large community. You could also just simply program the ESP 8266!  

The other option if you really desire to do a custom PCB is to use a WiFi enabled MCU such as the TI CC3120. These types of chips would throw a lot more hurdles at you in terms of soldering, programming, designing, troubleshooting, etc. But power to ya' if you wanna tackle that! Hopefully this helps.  

Feasibility

Of course it's possible for a single individual to take on this sort of project. I currently am! I'm using BLE instead of WiFi but I have the same goals. Large companies have teams of people who pack their devices with every feature you could need and make beautiful UIs and have UX specialists making the experience amazing. They just put a lot more time into their presentation. But, for a hobbyist, you only want to transmit sensor data which isn't nearly as monumental, or beautiful, of a task. I'm not as much into the programming side but you could most definitely achieve this if you want to put in the effort.