Is it generally best to book hotel stays with the corresponding card, e.g. Book a stay at Hilton with the HHonors Amex.

I have several CCs, and could probably have more. Would it strongly accelerate miles and points collection to have more cards....say the SPG AMEX card for all the SPG stays? and so forth?

You need to check what each card offers. If you have two choices to buy an airline ticket on say United and your choices are

Pay with an American branded card and get one American mile per dollar

or

Pay with a United branded card and get 3 miles per dollar (depends on the card)

you have to decide which miles are more important to you. Same thing with hotels, etc.

Remember though you need to get enough miles to actually use them (well at least I believe that's true, some people just like collecting) so if you pile up a few here and a few there and some here and some there you may never get enough anyplace to actually make it worth the effort. You need to balance the added rewards of another card against the chance you may not be able to use those miles later.

As a general rule, yes it is better value to book with the corresponding card, but it becomes useless unless you are able to hit the awards you want. If you travel a lot, then I'd recommend choosing the corresponding card, but if you travel lightly, I'd recommend finding a card that gives good value for all purchases.