ICP filing only applies if the server itself sits inside mainland China. If the machines you were quoted are in the US, Hong Kong or anywhere outside the mainland, there is nothing to file, and no way to file even if you wanted to, because the filing is tied to a licensed mainland provider.
What you lose without it is speed and certainty rather than access. Mainland visitors still reach the site, but everything crosses the border through the GFW, so latency swings, large assets crawl, and you cannot put a mainland CDN in front of it. Sites run like that for years, right up until they stop, and there is no warning when it happens.
If you genuinely need the filing, it takes a company registered in China plus hosting bought from a mainland provider. Foreign individuals generally cannot get one on their own. Most people who only want decent speed for mainland users skip it and put the site in Hong Kong or Singapore.
Where is the server you were quoted actually located, and are your mainland clients paying you inside China or from abroad? Those two answers change the recommendation completely.