I would not do that, if I were you. Weedy plants tend to have many propagation avenues. You might be disseminating seeds or turning stems and roots into cuttings, each capable of developing into a new plant. So one weed could turn into a hundred....by your own hand, yikes.

What about starting a compost pile with them, give them the chance to give something back for your effort?

If you do go over the pile with the mower - use the catcher and wear eye protection just in case you've gathered a stone or two.

Find out, ASAP, about your local pest weeds and see what you might have inherited. As your new garden has clearly been cared for, you could be lucky. However, your local bird life will be bringing you seeds from plants you don't want to see in the yard - ivy, or barberry, or worse. So the sooner you can recognise a pest the less hassle you'll have overall.

Beware of anything that spreads by runners. Dry them, then burn is about the most foolproof way.
Quarantine anything you buy from a plant store or get given. Either grow it for a while in an easy care part of your yard where you can quickly remove weeds or diseased plants. Or put them into pots until you've cleared the nearly-inevitable weedlings.

If you're allowed to in your area - a small garden incinerator can be useful for disposing of rose prunings and nasty weeds. You can spread the ash around the roses. They appreciate the minerals.