We give a new, strongly polynomial-time algorithm and improved analysis for the metric $s-t$ path TSP. It finds a tour of cost less than 1.53 times the optimum of the subtour elimination LP, while known examples show that 1.5 is a lower bound for the integrality gap. A key new idea is the deletion of some edges of Christofides' trees, which is then accompanied by novel arguments of the analysis: edge-deletion disconnects the trees, which are then partly reconnected by `parity correction'. We show that the arising `connectivity correction' can be achieved for a minor extra cost. On the one hand this algorithm and analysis extend previous tools such as the best-of-many Christofides algorithm. On the other hand, powerful new tools are solicited, such as a flow problem for analyzing the reconnection cost, and the construction of a set of more and more restrictive spanning trees, each of which can still be found by the greedy algorithm. We show that these trees can replace the convex combination of spanning trees in the best-of-may Christofides algorithm. These new methods lead to improving the integrality ratio and approximation guarantee below 1.53, as it is already sketched in the preliminary shortened version of this article that appeared in FOCS 2016. The algorithm and analysis have been significantly simplified in the current article, and details of proofs and explanations have been added.