What "cheap" bits are you using?
I thought I got the two individually from Amazon and see some available but looking just now I can't find the exact ones I got. I think they were $7-8 each but this was some years back. I also have the sets with a bunch of sizes but they don't go up to 2-1/6" or 2-1/8".
But as I mentioned, I no longer use Forstners to make recesses in face grain blanks. For bowls, platters, and other side grain pieces, I always start by holding the blank with a screw chuck. (Glazer, for me)
For end grain such as boxes and vessels, I do what others mentioned, drill a small depth hole, then hollow with a spindle gouge or often with one of the Hunter tools. I have a variety of Hunters with different shaft shapes and cutter sizes and can always find one that does the job. Make things so easy. I think John Lucas has some videos on the Hunter tools and how to use them.
And I mentioned this recently - when making one particular thing, the bell for a handbell ornament, the absolute best way I found to hollow is with a diamond parting tool! Might sound odd but really works well, controllable, precise, quick.
After hollowed to a thin wall thickness, I smooth the inside with either a Hunter tool or a spindle gouge. I cut this one in half to show how I make the wall thick at the center and thin at the rim.
I think I'm scheduled to do a demo on these in Chattanooga this fall.
JKJ