Yes, the "donut" tends to get narrower in the vertical plane as the gain of an "omni" goes up. However, this typically doesn't impact the ability of close-in stations to connect as the signal levels are much higher at closer ranges than they are at longer ranges. Unless you've got a very high gain "directional" antenna (such as a dish, reflector, or long-boom unit), you're not going to see any meaningful change in achievable modulation rates and bit-error rates for close-in units.
Beware of pigtails and feedlines. Anything you get from auction/import sites likely has significant loss, with pigtails alone able to negate the actual, if not claimed gain of a "rubber duckie" antenna. There are two common cable sizes for the uFL-style connectors, the smaller one is very lossy at 2.4 GHz, and even worse at 5 GHz. The 1.32 mm coax is notably better. Use of quality connectors is also important. Ordering quality pigtails from a reputable manufacturer through a recognized professional distributor is highly recommended.
Checking my orders, I have used https://www.mouser.com/ProductDetail/Taoglas/CAB721 with data at https://www.mouser.com/datasheet/2/398/CAB.721-4307.pdf in the past. Note that these quality 100 mm pigtails show 1 dB of loss. What you get from an auction/import site is unlikely to come close to that. (These may not be the right "gender" for your antennas as there is M, F, and reverse for both of those.)
Feed lines of any length, between the connectors and the cable losses themselves are very significant. Use of any cable that isn't of the quality of LMR240 or better is likely a net loss. Any of the typical "RG" series found on auction/import sites is pretty much guaranteed to be a net loss for you at 2.4 GHz, and even worse at 5 GHz.