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So 1♦-1♠-1NT-2♣-2♠ does indeed show exactly 3 card support for most people.
Note that after a 1nt rebid, it's common to use a gadget "new minor forcing", or various types of "checkback stayman" to get partner to show belated 3 cd major support. However these gadgets are not part of vanilla SAYC, and require special agreement. Without that agreement, simple non-jump new suits at the 2 level by responder after a 1nt rebid are by default natural and non-forcing, so would be a poor choice with 11-12 pts, as opener might pass with game on (more or less have to rebid 2nt, if not playing the gadgets). If possible make the "new minor forcing" (bidding in the minor partner didn't open) or "checkback stayman" (bidding 2c always artificial) or "2-way checkback" (both minor bids artificial, 2d game forcing) agreement with partner before the game, so you are on the same wavelength.
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So it is not widely accepted that SAYC (with or without useful extras) is a low grade bidding system?
The problem with SAYC is that it wasn't really constructed to be a good bidding system but rather a collection of commonly used treatments/conventions for individuals and tournaments where people were forced to use the same system. So there are a lot of inconsistencies & holes in spots. You can fix up these spots with more complete agreements & alterations here & there, but then you are really playing "Standard American" (the more generic term for the family of systems) with your own set of gadgets/treatments, and no longer playing "SAYC".