Funding and institutional policies are the major barriers to your plan.
There are a fairly large number of fellowships available for newly-minted PhDs looking to start a postdoc. These are often only open to people who graduated within the last 1-2 years. A number of other fellowships have slightly longer cut-offs: the Burrows-Wellcome Fund CASI award cuts off at 60 months post-PhD, and the NIH's K99 program is also limited to people within 4-5 years of their degree, and all training grants cut off at ~7 years. At the same time, many places also prevent non-"permanent" staff from applying for pure research grants, on the grounds that you're a) committing the university to do years of work but b) might not have a funded position then. (Yes, this seems to apply even when the grant contains funding for that very position. No, they don't seem to see the Catch-22).
At the same time, you're also getting more expensive. My institution has a 5 year "term limit" for postdocs, after which you have to become a "Research Associate" or "Research Assistant." This title change requires a slightly higher (though still objectively low) salary and more benefits (retirement, etc). The interaction between you a) costing more and b) definitely requiring money from a research grant, leads many lab heads to prefer younger applicants who cost less and could get their own money soon. To be clear, I think this is a totally ridiculous, wasteful, and possibly discriminatory—situation, and these policies ought to be reworked to avoid it. Nevertheless, it's going to be tough until they are.
The solution is to make sure that you're a good value for the money. Along with producing solid research on your own, take an active role in helping out with other projects in the group. Look for money where you can. Some foundations don't care much about career stage. You can also look for funding opportunities where you'll have a major role, even if you're not the formal PI. There are a very few staff scientist funding programs too. The NIH has an R50 "Research Specialist" grant that supports similar jobs (though with some restrictions on the person and environment), as does the Chan Zuckerberg Initiative.
Finally, you can lobby funders and agencies to reconsider their time-based eligibility limits. The NIH recently extended some K99 windows from 4 to 5 years for under-represented applicants, and I think the Wellcome Trust is removing all time considerations from some programs (though not CASI, which is probably the most relevant to a computer scientist).