^^ supported except that all my ITIL training has been driven by my employers in the past. But definately the larger ones have more scope to provide/support/enable it.
But frankly if you want consistent processes you need to select your frameworks and roll with it.
ITIL is one.
Other buzzwords will be PRINCE2, Agile vs Waterfall, SDLC.
But to center in on the OP and offer some useful advice, you need to get your foot in the door and show the right key competencies and a good attitude. A good employer will, if they have the capacity, take you on and grow you, if you can show yourself as worthwhile of that investment.