Is it possible to use a small cooler to do partial mash like you can for 5-10 gallon all grain? Everything is see says to use a kettle and bag. I understand this but why not use an insulated cooler for your grain rest?
Surely can, I actually used a 5 gallon cooler for my few partial mashes before I went to AG. Then I used the 5 gallon cooler for 3 gallon AG batches because I couldn't do a full 5 gallon on my stovetop. A 3 gallon cooler would be easier to keep your temps up for PM though, was a bit tough with a 5 gallon because it wasn't filled very full.
You can definitely use a smaller cooler for the mash.
Many use the kettle and bag, brew in a bag (BIAB) method as that method requires less equipment. Speaking for myself, I built a 10 gallon mash tun as I find it's more versatile in terms of different batch sizes than smaller coolers. I only wanted to build one cooler versus having multiple smaller coolers, but your situation might be different.
Northern brewer sells a 3 gallon cooler setup to do partial mash brews. Most people go with the brew in a bag method because it requires the least amount of extra equipment, just a grain bag. If you are planning on eventually stepping up to 5 or 10 gallon all grain batches then I would recommend the BIAB method because once you get to all grain you will no longer need the 3 gallon cooler. However if you are brewing in an apartment or someplace where space is limited and you aren't going to be going all grain anytime in the near future then going with a cooler might be a good option.
I have friends that use the partial mash BIAB method with very good results. I personally found it a bit of a PITA but I only tried it once before going to all grain. I would suggest giving it a try a few times to see if you can refine your process to get it to work for you. If you still find it a PITA then going with a cooler might be a good option. I have found mashing in a cooler very simple and easy as you don't have to worry about maintaining temps like you do when you use BIAB.