As suggested in the comments, I suppose this kind of filters depends on the audience of the website. HomeAway is oriented towards whole house rentals, so they would expect a kitchen comes along. However, like you, I would be glad to see that such websites have extensive filtering options for most people's needs. And they may be open to introduce more if you send them an email, or at least add a mention to the ads without any kitchen.
I looked a bit around and found a couple websites with such filters and covering a wide offer, including USA and international places.
AirBnb offers relatively many options (including kitchen and cabin) to filter the places you want to go to (even a First Aid Kit). Unlike HomeAway, I think AirBnb also targets those people wanting to rent a room and still being able to cook their own meals.
Booking.com offers also many filters including kitchen and kitchenette (check either one). Booking.com targets a super-wide audience with hotels, bed and breakfast and cottages/cabins.
Hostels.com as well as Hostelworld (same parent company, apparently same search results filtering) also have kitchen filters (and also a 'cooker' filter). Again the audience matters, since the hostels are also popular for the group atmosphere and cooking your own meal. These are rarely cottages or cabins, though.
Couchsurfing hosts would also be likely willing to share their kitchen, but again you will likely not be able to find a cottage.
In the end it seems to vary from website to website. There are a couple listed above that offer filters on kitchens.