A basic, ‘one-man’ brake bleeder is essentially a length of hose with a one-way valve to allow fluid out and prevent air being sucked back in. The very cheapest tyres have a bung in the end of the tube and a lengthways razor-slit just above it. They work fine, but after a year or so, the plastic hose hardens due to the chemical effect of the brake fluid and the ‘valve’ effect is lost. More expensive types have a proper ball-valve in the end, which should, in theory, last indefinitely.
Before starting to pump the pedal, fill the reservoir and open the bleed valves. Let gravity push the fluid through initially, before you connect the device. If the master cylinder is empty, ‘tickle’ the pedal with your foot and you should see a stream of bubbles come up through the reservoir as the cylinder bleeds itself. This is easier than trying to push the air all the way through to the bleed nipples.
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I think you meant to say cheapest "types", not tyres.