Impressions count really implies a view of a tweet, and there’s no way to get that information through the general API, let alone with no API. As you correctly discussed, followers count is not that meaningful, because some are inactive, some may just be not online and miss the tweet, etc.
I prefer to use the term “engagement” which is either a retweet or a like. I get both of those through the API. Without using an API, I feel your options are fairly limited.