On opening the file picker on a iPad you should be able to directly select a file.
On opening the file picker on a iPad you are not able to select any files. You do get a button (that you don't get in other browsers) to enable file selection (see screenshot). Since where using the file picker it's weird that the user has to click a button first to pick a file.

To reproduce you should open the filepicker on any iPad device. I used below configuration for the filepicker.
<script type="text/javascript" src="https://js.live.net/v7.2/OneDrive.js"></script>
<script type="text/javascript">
function launchOneDrivePicker(){
var odOptions = {
clientId: "***",
action: "query",
multiSelect: false,
advanced: {
filter: "",
scopes: ['https://graph.microsoft.com/User.Read', 'https://graph.microsoft.com/Files.ReadWrite', 'https://graph.microsoft.com/Sites.Read.All'],
isConsumerAccount: false,
endpointHint: "***"
},
success: (response) => {
console.log(response)
},
error: (response) => {
console.log(response)
},
cancel: () => {
}
}
OneDrive.open(odOptions);
}
</script>
<button onClick="launchOneDrivePicker()">Open from OneDrive</button>
Thank you for your contribution to OneDrive API Docs. We will be triaging your incoming issue as soon as possible.
Thanks for reporting this @kverhoef, this looks like an issue with the host page and perhaps special mobile behavior. We'll investigate and get back to you.
cc: @ThomasMichon @margoflynn
We have a fix in-flight for this @kverhoef. It was inadvertently caused by https://github.com/OfficeDev/office-ui-fabric-react/pull/8605. Thanks for the report.
I will check roll-out status and get back to you. Thanks to @ThomasMichon for the fix.
Most helpful comment
We have a fix in-flight for this @kverhoef. It was inadvertently caused by https://github.com/OfficeDev/office-ui-fabric-react/pull/8605. Thanks for the report.
I will check roll-out status and get back to you. Thanks to @ThomasMichon for the fix.