Does this decide who gives to whom?
No, and it would be the wrong tool for it. Assigning givers to receivers has to stay hidden from the person running it, which is exactly what a wheel everybody is watching cannot do — the whole room would see the pairing, including you.
Use a dedicated draw for the assignment, then use this for everything that happens in public afterwards. That is a real division of the job rather than a limitation: the part that must be secret and the part that must be seen want opposite tools.
What is the order actually for?
Opening. A Secret Santa of eight people with no order is eight people opening at once and nobody watching anybody, which is the one thing the format exists to avoid. Drawing the sequence takes ten seconds and gives every gift its moment.
It settles the smaller questions too: who picks first from a pile of unlabelled parcels, who goes next in a steal round, and which name buys in which week when a group is spreading the cost.
How do I run it when half the group is remote?
Press Share. The link opens the same wheel with the same names on anybody’s device, so the people in the room and the people on the call are looking at one list even if only one person spins.
For a video call, add ?embed=1 to the link: the page renders the wheel and the result alone, with no toolbar around it. That is what a shared tab or an OBS browser source wants — the wheel, full size, and nothing to explain.
What else is this wheel good for?
It is the name wheel with elimination on. If the parcels are numbered rather than named, the number wheel draws the number and the label stays on the gift.
Away from December the same draw is a presentation order. If you are drawing one winner from many rather than an order for everybody, that is a prize wheel.