joined as an HEO 12 years ago in my mid-20's on a big graduate recruitment drive, still an HEO! Most of my friends who joined at the same time and are similar age to me have moved up to SEO within the last couple of years. One or two have got G7 roles, and one is a G5 but she was ambitious and was a G7 within 4 years of joining.
I think EO is quite a tricky grade, in operational environments it can be a good grade to get lots of experience of case work roles with some autonomy, or you could be managing a team, but in other areas like policy it will be quite a junior support role. I found it tough when I moved departments a couple of years ago as operational work didn't translate over that well,and my new post seemed very high level in comparison with previous roles so I'm glad I moved to gain a different perspective.
I always advise new starters to get involved in extra corporate projects, staff networks etc to raise their profile and increase the range of skills you're practising, which will help develop your competencies (although as noted by a pp not sure how this will be measured with the new success profiles).
I would suggest scoping how likely new jobs at a higher grade are materialise in your current department, is there a restructure pending, is staff turnover high, these situations may give an opportunity to move up the ranks more quickly. If not then consider moving sideways to another department to get gain experience in a totally different role
if your face doesn't fit, move on! I've seen people who are perfectly capable get passed over for promotion because networking and having a good reputation is more likely to get you small projects, temporary promotion etc which gives you an edge for permanent roles.
I'm pretty happy at my level as I've always looked for a job rather than a grade, but I'm lucky in the North where I live my wage goes a long way.