Kind of a junior crypto question here, but does a the strength of a pub/private ecdsa key pair weaken if the privkey is used to sign a significant number of msgs.

I know that reusing the K value is instant death, but assuming the msgs were generated correctly should one be concerned never using a new receiving address?

I ask bc if one were to create a bitcoin service that wanted to provide an instant proof to a user of funds under administrator control, the user could send an arbitrary hash value to the server, which could then be signed by all the privkeys corresponding to all publicly controlled pubkeys.  

would the creator of such a service have to worry about a user DDOSing the service in order to scrape a large number of sigs generated by all his privkeys for all the loot he controls, as a way to make it easier for the attacker to derive the privkeys?

That is to say, imagine if an exchange operator did this.  Should he be worried about using the privkeys securing user funds to send perhaps millions or billions of sigs to an adversary trying to derive the privkeys?