Basically, has knowledge of metallurgy, manufacturing, and mechanics improved enough in that timespan that a cheaply-made sword now would surpass a well-made one from a long time ago?
The short answer is that the highest quality 13th century steel swords (which would typically be iron/steel laminated/composite construction rather than all steel, except for Indian/Central Asian/Middle Eastern crucible steel swords) would be better.
In terms of the quality of steel, they would be similar. Good bloomery steel (after folding repeatedly to get rid of slag and to homogenise carbon content) would be comparable to modern low-alloy steels (e.g., 1050, 1060, 1070, etc.). Some modern steels such as specialised spring steels and shock-resistant tool steels would provide additional toughness and/or wear resistance, but I assume your "modern low-grade steel" excludes those.
Where modern metallurgy wins is the amount of labour that is required for that steel. For pre-modern bloomery steel, smelting and forging could take about 10 hours per kg of steel (not counting labour costs of mining and charcoal production), while for modern steel, the labour costs are so low that wholesale prices are about US$1/kg.
Modern metallurgy also wins in terms of heat treatment. First, we can find out what is in our steel (e.g., using a mass spectrometer). This tells us what the optimal heat treatment is. Especially for tempering after quenching, this knowledge, along with temperature-controlled ovens, lets us aim for the best compromise between hardness and toughness. The difficulty of reliable tempering in pre-modern times, when the composition of the alloy was unknown, and accurate thermometers and thermostats were unavailable, made heat treatment much more hit-or-miss. However, the traditional solution of laminated/composite construction greatly simplifies the problem, and allows simple heat treatment to produce an effective sword with a hard edge and a tough body. It is quite possible for a traditionally-made sword, with traditional heat treatment, to have a harder edge than a modern mass-produced sword.
Where traditional sword-making wins over most modern mass-produced swords, is handling and balance, at least as far as the best swords go. In principle, modern mass-production using drop-forging (as used for very small numbers of modern swords) or roller-forging (as used for modern military sword production in the late 19th century and into the 20th century) can provide the same, but typically modern mass-production of swords uses stamping/cutting of sheet metal. In any case, even the best modern drop or roller forging won't provide anything better in terms of handling than the best of pre-modern swords.
Noting that many modern mass-produced swords are decorative wall-hangers, not intended for use. As such, they often use construction methods that are inappropriate for functional swords and would probably result in rapid failure in use, and use alloys better suited for decorative rather than functional swords. Excluding those from the comparison, one is left with
modern military swords, also not usually intended for actual use and using alloys and heat treatment more appropriate to dress swords than functional swords, and
modern functional historical replica swords.
Such replicas are rarely made using modern mass-production techniques. Hand-forging in the traditional way is common. Heat treatment of cheap replicas is often poor. Balance and handling are often atrocious compared to good swords. Where these swords win over the average original historical sword is that they start with good quality steels (i.e., your "modern low-grade steel"), either a low-alloy carbon steel or a spring steel like EN45/9260. However, this brings them level with the best old swords, and poor heat treatment and poor design can leave them greatly inferior.