The analogy is facile. Helium fuses at a temperature (10^8 K) roughly ten times higher than hydrogen (10^7 K), so a better analogy would be alcohol and thermite. That higher temperature is achieved only by massive gravitational contraction after hydrogen fusion [EDIT: in the core] is exhausted.

EDIT:

To expand, different mass stars undergo radically different life cycles, so "hydrogen fusion is exhausted" means different things for different stars. In all cases, fusion occurs only in a tiny core region.

For the lightest stars, convection (think rapidly boiling water) churns the entire star, so all of their hydrogen will eventually fuse. This will take much longer than the age of the universe, but even in the distant future, they will never compress enough to generate helium-fusing temperatures.

For heavier stars, including the Sun, convection only mixes the core region, so exhausting hydrogen fusion only means fusing all the hydrogen in the core. The tricky wrinkle is that after the exhaustion of core hydrogen, after the core collapses and heats, and after helium fusion starts, a thin shell of previously-pristine hydrogen surrounding the fusing helium can, itself, commence fusing as well. Hence the first, clarifying edit above.

The analogy is facile. Helium fuses at a temperature ($10^8\ \text{K}$) roughly ten times higher than hydrogen ($10^7\ \text{K}$), so a better analogy would be alcohol and thermite. That higher temperature is achieved only by massive gravitational contraction after hydrogen fusion [EDIT: in the core] is exhausted.

EDIT:

To expand, different mass stars undergo radically different life cycles, so "hydrogen fusion is exhausted" means different things for different stars. In all cases, fusion occurs only in a tiny core region.

For the lightest stars, convection (think rapidly boiling water) churns the entire star, so all of their hydrogen will eventually fuse. This will take much longer than the age of the universe, but even in the distant future, they will never compress enough to generate helium-fusing temperatures.

For heavier stars, including the Sun, convection only mixes the core region, so exhausting hydrogen fusion only means fusing all the hydrogen in the core. The tricky wrinkle is that after the exhaustion of core hydrogen, after the core collapses and heats, and after helium fusion starts, a thin shell of previously-pristine hydrogen surrounding the fusing helium can, itself, commence fusing as well. Hence the first, clarifying edit above.