Ok - this is a stupid question but I need a straight answer. If you take an omni antenna and mount it horizontally instead of vertically does that change the polarization? Short of all the obvious stuff about the sphere of coverage being wacked…
Omni in different settings, so for example V (normally) in horizontal position won’t be omni in horizontal in my understanding. For example the antenna has horizontal beamwidth 360degrees (as normally omni should have) and 8 degrees in vertical. When you change it (rotate of 90 degrees), you will have 360 deg V and 8 deg H Only panel antennas are reasonable to be rotated of 90 deg to change the polarization, there the H and V beamwidth are comparable.
If you need omni H, buy omni H, they are available on the market. Don’t invent “new” antennas
If you use a higher gain antenna then, as you said, you have a narrower beam. Lower gain antennas have much wider beam. Something like a 2dBi won’t really have a significant difference in coverage if turned sideways, however a 12dBi would.
It won’t change the gain, but it will totally alter the coverage area.