Support for early abortion is very different than for late term. Confusing one with the other is intentional.
If you trust the House and the Senate to not legalize late term abortions you must have missed how the Republicans who promised to repeal Obamacare only voted on that bill when they knew their vote would go nowhere. When their vote would have been signed into law by President Trump they suddenly wouldn't vote to repeal Obamacare.
If the federal government regulates abortion you can bet that in a few years that law will read like Planned Parenthood drafted it.