Not sure what you are talking about Verizon or AT&T's remote access, do you have an article about that? I use an android phone I bought from China (xiaomi phone, no Google Play), on AT&T's network, it works just fine. I'm not sure how AT&T can "remote access" my phone.
The network operators reserve the right to push new firmware to your phone/device. You did read the fine-print of your contract, right? This lets them brick phones with a specific IMEI if it appears to be mis-behaving.
Your Device must be compatible with, and not interfere with, our Services and must comply with all applicable laws, rules, and regulations. We may periodically program your Device remotely with system settings for roaming service, to direct your Device to use network services most appropriate for your typical usage, and other features that cannot be changed manually.
You agree that you won’t make any modifications to your Equipment or its programming to enable the Equipment to operate on any other system. AT&T may, at its sole and absolute discretion, modify the programming to enable the operation of the Equipment on other systems.
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AT&T Wireless Customer Agreement