To put it simple: with increase in bitcoin price the value of gas fee rises.
It is only true if fee size is the same.
No, it's not true in any case!
The fees depend only on network congestion. We have been at the same price levels 6 months ago and the fees were in the 1-2 satoshi per byte range, now it's 20x times more.
Even if OP is talking about CEX fees, there is no longer a thing of size since no exchange is charging by the size a withdrawal.
Also, there is no "gas" in Bitcoin!