Drilling Stainless is all about technique.
It work-hardens VERY easily.
"Moderate pressure" is almost certainly the problem.
Each pass of the cutting edge leaves a work-hardened layer beneath it. This is very thin and, when things are working well, the next pass of the cutting edge is through the unhardened layer just beneath.
When things go wrong, the drill doesn't move forward and the next pass of the drill doesn't cut, but slides over the work-hardened surface, work-hardening it more.
The drill will (often) become blunt and the problem will just get worse and worse until the operator gives up.
The only realistic way out of this that I know is to stop immediately the drill stops cutting, change for a new/sharp drill bit and start again, with enough feed pressure to cut through the work-hardened layer on the first pass.
Cobalt drills do seem to give a little more time between the first non-cutting pass and the drill becoming too blunt to get through the work-hardened layer on a restart, but it's not a big difference IME.
Cobalt is not a magic bullet. I suspect that a lot of the guys who know enough to use HSSCo drills are also the guys who know how to drill stainless (and stuff generally) properly.
I would not recommend using HSSCo to the OP. I think the fundamental problem is technique. HSSCo tends to be harder than HSS, but also more brittle. Adding (expensive) broken drills into the mix seems unlikely to help much.
I would recommend using new short (stub-length) drills, which are stiffer and easier to keep intact in small sizes, centre-punch the hole as emphatically as possible (to disrupt the work-hardened layer) and to drill in steps, starting with a 3/32" or 1/8" drill, low-to moderate speed and firm feed.