No, a class 10 slot needs at least a class 10 card to function properly. It won't work with a class 4 card, because the video is probably HD, which is faster than can be written to a class 4 card.
I have a few class 4 cards. I can and
have put them in a fast slot and they work. I have some relics in class 2 even. In class 4 my back stock ranges all the way up to 32GB A modern phone with an SDXC slot recognizes them and they read/write files. It's fine for documents and .mp3 if the bit rate isn't outrageously high. You can even shoot some higher res photos if you don't mind some camera lag while waiting for the card to catch up before you take the next shot.
Heck, I have one particular 8GB card in class 4 that outperforms some class 10 cards when I've done read/write tests on them.
The class is a minimum guarantee of speed. Some cards will outperform, but that's on a case by case basis.
It's not the slot that is lacking when it is class 10+ and the card is class 4. The card works, but it bottlenecks at high data rate tasks and performs slowly.
If I put a class 4 SD card in a fast slot there is no failure to see the card or use it - task depending. Yep, it won't do 4k video because it isn't fast enough. But it in no way causes the slot to become functionless as you seem to be implying.
"Properly" becomes relevant to what the user tasks the card to do. You seem to be limiting use to recording/playing back high bitrate video and the Op in no way indicated that is what they want to do.
Most phones in use today can handle SDXC cards, and the upper limit on that is 2TB, so we don't have to worry about size
The OP neglected to state what device he has, so we don't know if his slot is SDXC or SDHC or even (yes some of these are still on the street working hard) plain-jane SD. We have no gauge of what capacity their SD slot can handle.
Would I recommend a card of less than class 10? No. And with the EVO, as I said, I don't think we have to worry.
Does an SDXC slot rule out using a card less than class 10? We have to agree to disagree. My real world observations show otherwise since I rarely shoot video, stream high bitrate off a card and don't think I've ever recorded 4k except as a camera test. Less than class 10 is not optimal, but it still works.