First, I'm assuming these are HS seniors.
Second, you described (pretty much) the scenario of the HS pro scouts: a few gather and always talk amongst themselves (usually pointing out all the player's flaws); a player passing the eye test (with a radar gun/stop watch/video assisting) will get a questionnaire and/or a business card; and as the player's in the scout's area perform (i.e., demonstrate to the scout a prospective MLB tool) a home visit wil be scheduled.
Cross-checkers appear for prospects who could be considered higher round selections (lets's say 11-15) and also to evaluate players who need to be seen by even higher ups. If an asst GM or high execs of a club's amateur scouting start showing up, the player is being evaluated as a single digit selection.
BUT, at any point in the scouting process a kid can disappear from the prospective club's draft board - think absolute college commitment, out of the ballpark bonus demands, injury, performance, character issues, anything really.
While there are 1200 spots, i think less than 300 HS kids are drafted. And way more than 300 (or even 1200) are scouted amd evaluated.
Clubs hold draft meetings towards mid-May. All the club's scouts participate with the goal being a ranking of the prospects (the list has already elimimated many - some were not current prospects [but will develop into one during College], bonus, etc.). For many prospects, clubs will hold workouts (sometimes regionally, sometimes at the home park) where lots of the club's talent evaluators can observe and try to reach a consensus.
Clubs pretty much argree internally about the first day of the draft; by day three it cam be an internal brawl as area scouts advocate for their picks (scouts get small bonuses for picks they sign). For players (day 2 and 3), often a phone call precedes getting picked to make sure the player can be signed, whatever. Just getting a call is meaningless, however, until the selection. (My son was called by half a dozen clubs during one of his drafts.)
It's a process with lots of nuance; just because a player is NOT drafted is not necessarily a reflection on his baseball tools; drafting a HS player has more puzzle pieces than drafting a college junior or a senior (which has 2 pieces: MLB tool + a steeply discounted bonus).