President Biden is attending the G7 Summit today. Here is his schedule

As POTUS, Biden has other leadership duties while on a State visit. A FLOTUS engaging in other activities isn't unusual. Spicer is a member of the GOP which is politically opposed to Democrats and Biden in general. More about perception of Biden's public conduct:

The UK is really not messing around with Joe Biden. Yesterday Boris kept his son away from Joe but let Jill meet him. Today Kate Middleton is touring Jill around a school but didn’t bring Joe. I guess they said they’re not going to have some creepy old man sniffing the kids!