A normal debugger uses documented API/syscall/instruction to look at state of a process it is permissionedpermitted to access. Just being a debugger alone cannot bypass OS memory protection, otherwise anyone who knows how to download stuff can gain admin access on any system.
Meltdown and spectre attacks use (previously) undocumented and unintended side-effects of certain internal processor design flaws to exfiltrate information that the attacking code are not permissionedpermitted to access.
In addition, these attacks (Meltdown in particular) are not very good for debugging purposes as they can only passively observe memory on a probabilistic level. A deliberate backdoor used for debugging, e.g. JTAG on embedded systems, would allow direct and realtime memory access.