I'm a civil servant, it isn't the idea that progression isn't easy it's just clarifying to some that it isn't as easy as they might imagine in that it isn't guaranteed. However, if you're willing to work hard, develop yourself and expand your experience then there's no reason why you couldn't progress.
2-3 years is a realistic timeframe to gain new experience in the HO role before applying for SEO jobs. Personally I was an AO up until January before coming an EO. I've been in the EO role for 4 months before being offered an HEO position. Whilst that wouldn't be seen as a regular occurrence I was able to get the HEO role by using the skills I had learnt as an EO.
Personally I think the key to developing yourself in the civil service is taking on responsibility for areas of work that wouldn't fall under your direct remit or getting involved in things outside your usual role. Things such as working groups or volunteering to be the site point of contact for a particular project that's going on regionally. They offer different experiences to that in your day to day work. Volunteer for everything you can.
These case working roles sound like they would offer great experience in decision making, collaborative working, dealing with confrontation and therefore people skills but there is also no direct line management responsibilities. From SEO grade and above a lot of operations roles are all about managing people and their performance, you may need to develop those skills in order to be successful in gaining promotion to the next level. That's just one example of personal development, it would all depend on which sort of roles you would be interested in moving into in the future.
I previously worked as a decision maker for the Home Office which gave me valuable decision making and collaborative working skills but I had never managed staff. Because of this I applied for a team leader role in DWP on promotion to gain that experience to help me move up again later on.
Civil service wide there are tons of vacancies coming up all the time from HEO level upwards so the opportunities are available for those with the required skills.
In terms of job security, in years gone by a civil service job was considered as a job for life but now it's the same as anywhere else. However, in 8 years I've never been anywhere that has had a compulsory redundancy package, I've seen a couple of voluntary packages but compulsory is almost unheard of. Senior managers would use it as a last resort. If redundancies were ever raised you're still able to apply for other civil service vacancies to escape that threat and if the threat of redundancy is real you would be classed as a priority mover so that if you did apply for other jobs you would have priority for it because your job is at risk. The only people I have known accept redundancy are those who genuinely wanted to.