Hi Jason,
Attempting to send a challenge can be pretty annoying, given how simple the operation should be when stuff like this happens. I assure you that you are not the only one; this isn't rare (Many threads like yours have been posted on the site with almost no solutions, unfortunately). Texting/emailing or, best of all, the messaging feature on chess.com makes the communication part more manageable. Lichess doesn't seem to have this problem.
This is what I usually do for Chess.com:
1) I send a message to a friend. They usually reply; If not, onto the next friend!
2) I message them, "I'll send you a challenge, and if you don't see it, feel free to send me a challenge!" This works every time, though I have not done it in a while.
(Then I go to my homepage to wait for the challenge. If that doesn't work I wait on the play page)
I hope this helps, and feel free to ask any questions if I evaded any aspects you wish answered. Happy New Year!
I don't think I can use the challenge feature. Perhaps someone can help me. But it might just be my account. Probably I am just not allowed to challenge anyone. Anyway, my guess is that there are different ways to challenge a friend to a game. None of them work for me, which is fine, but I wanted to see how others use the feature. For me, first of all someone is either on the site on not. If on the site then they are probably playing a game. I get this message for that situation, which is good and somewhat expected:

Sort of. I mean, it should send an email to the person if they are offline, alerting them I want to play. And if they are playing they should get a note I'm next. Right? Am I on another planet? I know I am, so that's a retorical question. Sorry.
Moving on, if the person I want to play is actually online and just done playing, I get this message:
Seems reasonable enough except I can always play regular games with 4 bars. I never have any connection issues. I mean I have gigabit internet from the most expensive ISP (verizon fios) using a fiber optic cable to the internet.
Any other options. Yes. I can at least get a link and send a message, right? Wrong. It causes my account to be muted. As a support person wrote,
I'm with you on this.
I read your message and I must say I understand what you mean. I too would want a way to know why my account was restricted if I was in your place.
Now, I checked your account again and I was able to find that the message you sent:
"https://play.chess.com/NyhXw"
Triggered our "spam filter". I understand that you were only sending a challenge to a friend based on your message, but unfortunately, our system is an AI so it can't tell when someone is sending links under a link that redirects us outside of the site (if that makes sense) or if it's someone being abusive. So, we have to kind of make it affect everyone equally until AI's get good enough to tell the difference, and by then they probably will have taken over the world, right?
OK, the bot can't tell if it's a url to this site itself. Seems strange. Whatever.
My question to anyone reading this strange post is how do normal people use the challenge feature? I realize it's not an option for me, but can any kind soul please share how it works for them? I'm just curious. Like perhaps I'm overlooking something simple.
Many thanks in advance for any attempt to help!
Jason