The "Manage Bot" button on a bot's profile deep-links to @BotFather with
start=<bot username>. parseBotCommand dropped the argument, so /start <bot>
just replied with the generic greeting instead of the per-bot menu.
Route "/start <arg>" to the bot's "What do you want to do?" screen (same as
/mybots then tapping the bot) when <arg> names one of the sender's own bots by
username or id; empty or unknown args keep the greeting.