In terms of founding a company and hiring a few people who would help at first, it’s possible to run a software company alone. There are tons of freelancers and “one-man show” companies that provide professional software services.
When you hire the first few people on your team, it’s also normal that you are nurturing and handling their work and helping out with onboarding, training, code reviews, and other activities provided by your non-technical hires (if any, at first).
As your company grows, it will be physically impossible for you to handle all communications internally with your team, sales pitches, writing proposals, doing customer support, providing other services that you are not familiar with (or experienced in), and so on. You will need to learn to delegate and identify leading roles within your team or start looking for external help.
From my experience, hiring management staff outside of the company is challenging and often hard to assimilate with team members. Sure, it’s possible and often preferable if there are suitable candidates with outstanding backgrounds in a similar role who can sort out all internal issues and scale your process as you grow. During the first few dozen hires, you may try to prepare some of your team members to step into more leading roles – senior engineers or architects, team leaders, technical project managers, and so on.
If you manage over one hundred people, unless you hire extraordinary team of rockstars who can self-manage and be proactive and participate in all client conversations, planning, estimates, and the like, your team will probably require some senior management staff that can deal with coordination, delegation, validating results, and managing expectations. This could still be somewhat flat (only one tier between you and the rest of your employees), or hierarchical with multiple branches and different management seniority.
Whenever you have an entrusted member who can take on an entire department (marketing, creative, operations, sales), you can consider C-level roles within your team or other tempting managerial stunts. Otherwise, bringing a formal partner into your organization with shared responsibilities would be a good alternative as well.
Things get more complicated if you look for funding and bring investors on board. The entire structure could be decomposed and rebuilt from scratch for higher performance results and better internal operations.