I believe your friend is being honest with you and wants to keep the friendship.

I was purposefully bad at texting someone back who I didn’t want to be friends with and when she got mad at me for it/said to me she was ending the friendship, I was relieved and let her go. I actually disliked her for thinking I owed her immediate responses and attention. No way would I have been begging for her friendship, even for appearances.

Sometimes I can be genuinely bad at texting with great friends/family, though. If I’m feeling socially exhausted/don’t want to engage. Maybe I only have energy for one catch-up that week. Mental health is hard, and if I were you I would try to be more understanding your friend might be going through a rough patch or this is just how they operate.

Phone technology can make us think everyone owes us immediate responses but sometimes people like to switch off/respond when they’re ready.

You’ve raised it, your friend has given a reasonable /presumably honest explanation and expressed they do value you, so I wouldn’t push it or it can be a big turn-off.