Nowadays it really depends on how the immobilizer works to figure out how to/if you can override it easily.

I'm familiar with a couple designs - and on my vehicle I found it trivial to temporarily override (with the wiring diagram in the service manual).

My vehicles are on the older end of having immobilizers, but on them automakers certainly didn't integrate it tightly with other systems. It is controlled by one of the main ECUs - but the lockout method is subpar.

Better integration of the immobilizer goes along with the development of new engine management systems - so there may be more integration on vehicles with the newest engine designs (but not necessarily if the engine descended from a previous design and they kept the engine management system mostly intact....).

Some immobilizers disable starting, some disable fuel pump, some disable ignition, and some disable injectors. The how these are electrically disabled also matters a lot.

It's still trivial to override the first two methods, and sometimes the second two, but at least the last two have the potential to require more than a wire to disable if they are better integrated with the ECU.