As a website developer I see lots of clients who have only been granted limited access to their own websites. I believe that the Developers do this to stop the clients uploading ridiculously large images or installing plugins that could break the site etc. This is your website though and is an important business asset. With a set of technical guidelines (image sizes etc) you should be able to update the site if you choose.
The SEO aspect of the site is complex and any changes to the site should be made with this in mind. Good SEO is inseparable from good user experience, so if there’s information about your business you need your clients or customers to know, it’s important for them – and for your SEO – that you’re able to add it. I would recommend chatting to your web developer or SEO team before you add new pages or radically change content, so that you can gain their insight into how best to integrate the new content.
Moving the site to new hosting won’t necessarily affect the SEO but do be aware that the quality of the hosting does affect the speed of the site, so choose a host that is appropriate for your site size and CMS, and has a reputation for reliability and good support.
Whoever manages your website should be open to conversations about how best to grow the site to match the growth of your business. If they are not then I’d recommend finding someone who is.